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Absolute Value Inequalities Notes

The document discusses absolute value, absolute value equations, and absolute value inequalities. It defines absolute value as the distance from zero on the number line, which is always positive. Absolute value equations account for both positive and negative cases, while absolute value inequalities split into two cases - one for positive values and one for negative values. An example shows how to write the inequalities for both cases of |x| < 4, determine the solution is an intersection of the cases, and represent the solution set as {x | -4 < x < 4}.

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Absolute Value Inequalities
What is Absolute Value?
What is Absolute Value? • Absolute Value is the distance from zero on a number line.
What is Absolute Value? • Absolute Value is the distance from zero on a number line. • Distance is always positive.
What is Absolute Value? • Absolute Value is the distance from zero on a number line. • Distance is always positive. • Symbol is | |.
What is Absolute Value? • Absolute Value is the distance from zero on a number line. • Distance is always positive. • Symbol is | |. • Example 1: |3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
What is Absolute Value? • Absolute Value is the distance from zero on a number line. • Distance is always positive. • Symbol is | |. • Example 1: |3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
What is Absolute Value? • Absolute Value is the distance from zero on a number line. • Distance is always positive. • Symbol is | |. • Example 1: |3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Example 2: |-3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
What is Absolute Value? • Absolute Value is the distance from zero on a number line. • Distance is always positive. • Symbol is | |. • Example 1: |3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Example 2: |-3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
Absolute Value Equation
Absolute Value Equation • Because the distance can be towards the positive or negative direction, Absolute Value Equations must account for both cases.
Absolute Value Equation • Because the distance can be towards the positive or negative direction, Absolute Value Equations must account for both cases. • Example 3: |x| = 4
Absolute Value Equation • Because the distance can be towards the positive or negative direction, Absolute Value Equations must account for both cases. • Example 3: |x| = 4 x = 4 or -5 -4 -3 -2 -1 0 1 2 3 4 5
Absolute Value Equation • Because the distance can be towards the positive or negative direction, Absolute Value Equations must account for both cases. • Example 3: |x| = 4 x = 4 or -5 -4 -3 -2 -1 0 1 2 3 4 5 -(x) = 4
Absolute Value Equation • Because the distance can be towards the positive or negative direction, Absolute Value Equations must account for both cases. • Example 3: |x| = 4 x = 4 or -5 -4 -3 -2 -1 0 1 2 3 4 5 -(x) = 4 x = -4 -5 -4 -3 -2 -1 0 1 2 3 4 5
Absolute Value Inequality
Absolute Value Inequality • How does Absolute Value work for Inequalities?
Absolute Value Inequality • How does Absolute Value work for Inequalities? • Example 4: |x| < 4
Absolute Value Inequality • How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve.
Absolute Value Inequality • How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve. Positive Case
Absolute Value Inequality • How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve. Positive Case Negative Case
Absolute Value Inequality • How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve. Positive Case Negative Case x<4 -(x) < 4
Absolute Value Inequality • How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve. Positive Case Negative Case x<4 -(x) < 4 x > -4
Absolute Value Inequality • How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve. Positive Case Negative Case x<4 -(x) < 4 x > -4 Remember to reverse the inequality when dividing by a negative.
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 4 (con’t) |x| < 4 • Graph the solutions. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 4 (con’t) |x| < 4 • Graph the solutions. x<4 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 4 (con’t) |x| < 4 • Graph the solutions. x<4 x > -4 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 4 (con’t) |x| < 4 • Graph the solutions. x<4 x > -4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Is the solution is an intersection (∩) or a union (∪)?
Example 4 (con’t) |x| < 4 • Graph the solutions. x<4 x > -4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Is the solution is an intersection (∩) or a union (∪)? • Check numbers in the original inequality to help you decide.
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities.
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺ |0| < 4 True, so the area between -4 and 4 is a solution.
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺ |0| < 4 True, so the area between -4 and 4 is a solution. • Check -5.
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺ |0| < 4 True, so the area between -4 and 4 is a solution. • Check -5. |-5| < 4 False, so the area less than -4 is not a solution.
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺ |0| < 4 True, so the area between -4 and 4 is a solution. • Check -5. |-5| < 4 False, so the area less than -4 is not a solution. • Check 5.
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺ |0| < 4 True, so the area between -4 and 4 is a solution. • Check -5. |-5| < 4 False, so the area less than -4 is not a solution. • Check 5. |5| < 4 False, so the area greater than 4 is not a solution.
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because the only number that worked was between -4 and 4, the solution is an intersection.
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because the only number that worked was between -4 and 4, the solution is an intersection. • The solution can be written as
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because the only number that worked was between -4 and 4, the solution is an intersection. • The solution can be written as {x | -4 < x < 4}
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because the only number that worked was between -4 and 4, the solution is an intersection. • The solution can be written as {x | -4 < x < 4} or as
Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because the only number that worked was between -4 and 4, the solution is an intersection. • The solution can be written as {x | -4 < x < 4} or as {x | x > -4 ∩ x < 4}
Example 5: What about |x| > 3?
Example 5: What about |x| > 3? Write the inequalities for the 2 cases and solve.
Example 5: What about |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case
Example 5: What about |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3
Example 5: What about |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3 x < -3
Example 5: What about |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3 x < -3 Remember to reverse the inequality when dividing by a negative.
Example 5: What about |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3 x < -3 Remember to reverse the inequality when dividing by a negative. • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 5: What about |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3 x < -3 Remember to reverse the inequality when dividing by a negative. • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 5: What about |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3 x < -3 Remember to reverse the inequality when dividing by a negative. • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality.
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺ |0| > 3 False, so the area between -4 and 4 is not a solution.
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺ |0| > 3 False, so the area between -4 and 4 is not a solution. • Check -5.
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺ |0| > 3 False, so the area between -4 and 4 is not a solution. • Check -5. |-5| > 3 True, so the area less than -4 is a solution.
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺ |0| > 3 False, so the area between -4 and 4 is not a solution. • Check -5. |-5| > 3 True, so the area less than -4 is a solution. • Check 5.
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺ |0| > 3 False, so the area between -4 and 4 is not a solution. • Check -5. |-5| > 3 True, so the area less than -4 is a solution. • Check 5. |5| > 3 True, so the area greater than 4 is a solution.
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because a number worked from both inequalities, the solution is an union.
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because a number worked from both inequalities, the solution is an union. • The solution can be written as
Example 5 (con’t): |x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because a number worked from both inequalities, the solution is an union. • The solution can be written as {x | x < -3 ∪ x > 3}
Example 6: What about |x| < -3?
Example 6: What about |x| < -3? Write the inequalities for the 2 cases and solve.
Example 6: What about |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case
Example 6: What about |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x < -3 -(x) < -3
Example 6: What about |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x < -3 -(x) < -3 x>3
Example 6: What about |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x < -3 -(x) < -3 x>3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 6: What about |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x < -3 -(x) < -3 x>3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 6: What about |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x < -3 -(x) < -3 x>3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 6 (con’t): |x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 6 (con’t): |x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • It looks a lot like Example 5 so far. Time for our check. Let’s check -4 first.
Example 6 (con’t): |x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • It looks a lot like Example 5 so far. Time for our check. Let’s check -4 first. ? − ( −4 ) <− 3 4 < −3 4 < −3
Example 6 (con’t): |x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • It looks a lot like Example 5 so far. Time for our check. Let’s check -4 first. ? − ( −4 ) <− 3 • What?! This isn’t true! 4 < −3 4 < −3
Example 6 (con’t): |x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • It looks a lot like Example 5 so far. Time for our check. Let’s check -4 first. ? − ( −4 ) <− 3 • What?! This isn’t true! 4 < −3 • Absolute Value is distance. Distance is 4 < −3 never negative. The Absolute Value can never result in a negative number.
Example 6 (con’t): |x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • It looks a lot like Example 5 so far. Time for our check. Let’s check -4 first. ? − ( −4 ) <− 3 • What?! This isn’t true! 4 < −3 • Absolute Value is distance. Distance is 4 < −3 never negative. The Absolute Value can never result in a negative number. No Solution
Example 7: What about |x| > -3?
Example 7: What about |x| > -3? Write the inequalities for the 2 cases and solve.
Example 7: What about |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case
Example 7: What about |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x > -3 -(x) > -3
Example 7: What about |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x > -3 -(x) > -3 x<3
Example 7: What about |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x > -3 -(x) > -3 x<3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 7: What about |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x > -3 -(x) > -3 x<3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 7: What about |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x > -3 -(x) > -3 x<3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
Example 7 (con’t): |x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph?
Example 7 (con’t): |x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? • Check 0.
Example 7 (con’t): |x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3
Example 7 (con’t): |x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 • Looks good. Try -5.
Example 7 (con’t): |x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 ? −5 >− 3 • Looks good. Try -5. 5 > −3
Example 7 (con’t): |x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 ? −5 >− 3 • Looks good. Try -5. 5 > −3 • Also looks good.
Example 7 (con’t): |x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 ? −5 >− 3 • Looks good. Try -5. 5 > −3 • Also looks good. • Is there a number that will not work in the original problem?
Example 7 (con’t): |x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 ? −5 >− 3 • Looks good. Try -5. 5 > −3 • Also looks good. • Is there a number that will not work in the original problem? • No, which means the solution can be any real number.
Example 7 (con’t): |x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 ? −5 >− 3 • Looks good. Try -5. 5 > −3 • Also looks good. • Is there a number that will not work in the original problem? • No, which means the solution can be any real number. All Real Numbers

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Absolute Value Inequalities Notes

  • 1.
  • 2.
  • 3.
    What is AbsoluteValue? • Absolute Value is the distance from zero on a number line.
  • 4.
    What is AbsoluteValue? • Absolute Value is the distance from zero on a number line. • Distance is always positive.
  • 5.
    What is AbsoluteValue? • Absolute Value is the distance from zero on a number line. • Distance is always positive. • Symbol is | |.
  • 6.
    What is AbsoluteValue? • Absolute Value is the distance from zero on a number line. • Distance is always positive. • Symbol is | |. • Example 1: |3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 7.
    What is AbsoluteValue? • Absolute Value is the distance from zero on a number line. • Distance is always positive. • Symbol is | |. • Example 1: |3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 8.
    What is AbsoluteValue? • Absolute Value is the distance from zero on a number line. • Distance is always positive. • Symbol is | |. • Example 1: |3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Example 2: |-3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 9.
    What is AbsoluteValue? • Absolute Value is the distance from zero on a number line. • Distance is always positive. • Symbol is | |. • Example 1: |3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Example 2: |-3| = 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 10.
  • 11.
    Absolute Value Equation •Because the distance can be towards the positive or negative direction, Absolute Value Equations must account for both cases.
  • 12.
    Absolute Value Equation •Because the distance can be towards the positive or negative direction, Absolute Value Equations must account for both cases. • Example 3: |x| = 4
  • 13.
    Absolute Value Equation •Because the distance can be towards the positive or negative direction, Absolute Value Equations must account for both cases. • Example 3: |x| = 4 x = 4 or -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 14.
    Absolute Value Equation •Because the distance can be towards the positive or negative direction, Absolute Value Equations must account for both cases. • Example 3: |x| = 4 x = 4 or -5 -4 -3 -2 -1 0 1 2 3 4 5 -(x) = 4
  • 15.
    Absolute Value Equation •Because the distance can be towards the positive or negative direction, Absolute Value Equations must account for both cases. • Example 3: |x| = 4 x = 4 or -5 -4 -3 -2 -1 0 1 2 3 4 5 -(x) = 4 x = -4 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 16.
  • 17.
    Absolute Value Inequality •How does Absolute Value work for Inequalities?
  • 18.
    Absolute Value Inequality •How does Absolute Value work for Inequalities? • Example 4: |x| < 4
  • 19.
    Absolute Value Inequality •How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve.
  • 20.
    Absolute Value Inequality •How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve. Positive Case
  • 21.
    Absolute Value Inequality •How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve. Positive Case Negative Case
  • 22.
    Absolute Value Inequality •How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve. Positive Case Negative Case x<4 -(x) < 4
  • 23.
    Absolute Value Inequality •How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve. Positive Case Negative Case x<4 -(x) < 4 x > -4
  • 24.
    Absolute Value Inequality •How does Absolute Value work for Inequalities? • Example 4: |x| < 4 Write the inequalities for the 2 cases and solve. Positive Case Negative Case x<4 -(x) < 4 x > -4 Remember to reverse the inequality when dividing by a negative.
  • 25.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 26.
    Example 4 (con’t) |x| < 4 • Graph the solutions. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 27.
    Example 4 (con’t) |x| < 4 • Graph the solutions. x<4 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 28.
    Example 4 (con’t) |x| < 4 • Graph the solutions. x<4 x > -4 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 29.
    Example 4 (con’t) |x| < 4 • Graph the solutions. x<4 x > -4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Is the solution is an intersection (∩) or a union (∪)?
  • 30.
    Example 4 (con’t) |x| < 4 • Graph the solutions. x<4 x > -4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Is the solution is an intersection (∩) or a union (∪)? • Check numbers in the original inequality to help you decide.
  • 31.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 32.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities.
  • 33.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺
  • 34.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺ |0| < 4 True, so the area between -4 and 4 is a solution.
  • 35.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺ |0| < 4 True, so the area between -4 and 4 is a solution. • Check -5.
  • 36.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺ |0| < 4 True, so the area between -4 and 4 is a solution. • Check -5. |-5| < 4 False, so the area less than -4 is not a solution.
  • 37.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺ |0| < 4 True, so the area between -4 and 4 is a solution. • Check -5. |-5| < 4 False, so the area less than -4 is not a solution. • Check 5.
  • 38.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x < 4, one that solves x > -4, and one that solves both inequalities. • Check 0 first. Why zero? Because it’s easy. ☺ |0| < 4 True, so the area between -4 and 4 is a solution. • Check -5. |-5| < 4 False, so the area less than -4 is not a solution. • Check 5. |5| < 4 False, so the area greater than 4 is not a solution.
  • 39.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because the only number that worked was between -4 and 4, the solution is an intersection.
  • 40.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because the only number that worked was between -4 and 4, the solution is an intersection. • The solution can be written as
  • 41.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because the only number that worked was between -4 and 4, the solution is an intersection. • The solution can be written as {x | -4 < x < 4}
  • 42.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because the only number that worked was between -4 and 4, the solution is an intersection. • The solution can be written as {x | -4 < x < 4} or as
  • 43.
    Example 4 (con’t) |x| < 4 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because the only number that worked was between -4 and 4, the solution is an intersection. • The solution can be written as {x | -4 < x < 4} or as {x | x > -4 ∩ x < 4}
  • 44.
    Example 5: Whatabout |x| > 3?
  • 45.
    Example 5: Whatabout |x| > 3? Write the inequalities for the 2 cases and solve.
  • 46.
    Example 5: Whatabout |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case
  • 47.
    Example 5: Whatabout |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3
  • 48.
    Example 5: Whatabout |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3 x < -3
  • 49.
    Example 5: Whatabout |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3 x < -3 Remember to reverse the inequality when dividing by a negative.
  • 50.
    Example 5: Whatabout |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3 x < -3 Remember to reverse the inequality when dividing by a negative. • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 51.
    Example 5: Whatabout |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3 x < -3 Remember to reverse the inequality when dividing by a negative. • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 52.
    Example 5: Whatabout |x| > 3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x>3 -(x) > 3 x < -3 Remember to reverse the inequality when dividing by a negative. • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 53.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 54.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality.
  • 55.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺
  • 56.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺ |0| > 3 False, so the area between -4 and 4 is not a solution.
  • 57.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺ |0| > 3 False, so the area between -4 and 4 is not a solution. • Check -5.
  • 58.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺ |0| > 3 False, so the area between -4 and 4 is not a solution. • Check -5. |-5| > 3 True, so the area less than -4 is a solution.
  • 59.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺ |0| > 3 False, so the area between -4 and 4 is not a solution. • Check -5. |-5| > 3 True, so the area less than -4 is a solution. • Check 5.
  • 60.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Check 3 numbers. One that solves x > 3, one that solves x < 3, and one that neither inequality. • Check 0 first. Why zero? Because it’s easy. ☺ |0| > 3 False, so the area between -4 and 4 is not a solution. • Check -5. |-5| > 3 True, so the area less than -4 is a solution. • Check 5. |5| > 3 True, so the area greater than 4 is a solution.
  • 61.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 62.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because a number worked from both inequalities, the solution is an union.
  • 63.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because a number worked from both inequalities, the solution is an union. • The solution can be written as
  • 64.
    Example 5 (con’t):|x| > 3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • Because a number worked from both inequalities, the solution is an union. • The solution can be written as {x | x < -3 ∪ x > 3}
  • 65.
    Example 6: Whatabout |x| < -3?
  • 66.
    Example 6: Whatabout |x| < -3? Write the inequalities for the 2 cases and solve.
  • 67.
    Example 6: Whatabout |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case
  • 68.
    Example 6: Whatabout |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x < -3 -(x) < -3
  • 69.
    Example 6: Whatabout |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x < -3 -(x) < -3 x>3
  • 70.
    Example 6: Whatabout |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x < -3 -(x) < -3 x>3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 71.
    Example 6: Whatabout |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x < -3 -(x) < -3 x>3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 72.
    Example 6: Whatabout |x| < -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x < -3 -(x) < -3 x>3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 73.
    Example 6 (con’t):|x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 74.
    Example 6 (con’t):|x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • It looks a lot like Example 5 so far. Time for our check. Let’s check -4 first.
  • 75.
    Example 6 (con’t):|x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • It looks a lot like Example 5 so far. Time for our check. Let’s check -4 first. ? − ( −4 ) <− 3 4 < −3 4 < −3
  • 76.
    Example 6 (con’t):|x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • It looks a lot like Example 5 so far. Time for our check. Let’s check -4 first. ? − ( −4 ) <− 3 • What?! This isn’t true! 4 < −3 4 < −3
  • 77.
    Example 6 (con’t):|x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • It looks a lot like Example 5 so far. Time for our check. Let’s check -4 first. ? − ( −4 ) <− 3 • What?! This isn’t true! 4 < −3 • Absolute Value is distance. Distance is 4 < −3 never negative. The Absolute Value can never result in a negative number.
  • 78.
    Example 6 (con’t):|x| < -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • It looks a lot like Example 5 so far. Time for our check. Let’s check -4 first. ? − ( −4 ) <− 3 • What?! This isn’t true! 4 < −3 • Absolute Value is distance. Distance is 4 < −3 never negative. The Absolute Value can never result in a negative number. No Solution
  • 79.
    Example 7: Whatabout |x| > -3?
  • 80.
    Example 7: Whatabout |x| > -3? Write the inequalities for the 2 cases and solve.
  • 81.
    Example 7: Whatabout |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case
  • 82.
    Example 7: Whatabout |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x > -3 -(x) > -3
  • 83.
    Example 7: Whatabout |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x > -3 -(x) > -3 x<3
  • 84.
    Example 7: Whatabout |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x > -3 -(x) > -3 x<3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 85.
    Example 7: Whatabout |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x > -3 -(x) > -3 x<3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 86.
    Example 7: Whatabout |x| > -3? Write the inequalities for the 2 cases and solve. Positive Case Negative Case x > -3 -(x) > -3 x<3 • Graph. -5 -4 -3 -2 -1 0 1 2 3 4 5
  • 87.
    Example 7 (con’t):|x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph?
  • 88.
    Example 7 (con’t):|x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? • Check 0.
  • 89.
    Example 7 (con’t):|x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3
  • 90.
    Example 7 (con’t):|x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 • Looks good. Try -5.
  • 91.
    Example 7 (con’t):|x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 ? −5 >− 3 • Looks good. Try -5. 5 > −3
  • 92.
    Example 7 (con’t):|x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 ? −5 >− 3 • Looks good. Try -5. 5 > −3 • Also looks good.
  • 93.
    Example 7 (con’t):|x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 ? −5 >− 3 • Looks good. Try -5. 5 > −3 • Also looks good. • Is there a number that will not work in the original problem?
  • 94.
    Example 7 (con’t):|x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 ? −5 >− 3 • Looks good. Try -5. 5 > −3 • Also looks good. • Is there a number that will not work in the original problem? • No, which means the solution can be any real number.
  • 95.
    Example 7 (con’t):|x| > -3 -5 -4 -3 -2 -1 0 1 2 3 4 5 • What is different about the graph? ? 0 >− 3 • Check 0. 0 > −3 ? −5 >− 3 • Looks good. Try -5. 5 > −3 • Also looks good. • Is there a number that will not work in the original problem? • No, which means the solution can be any real number. All Real Numbers

Editor's Notes