Solving linear inequalities in one variable

Solving linear inequalities in one variable

• 1.
  21/09/25 Solving Inequalities Key words Solve,Unknown, Inverse, Opposite, Inequality, More than, Greater than, Less than OBJECTIVE Solve Algebraic Inequalities OUTCOMES I know what inequalities are I can represent inequalities on a number line I can solve inequalities I can represent inequalities on a graph
• 2.
  What are inequalities? <, > , ≥ , ≤
• 3.
  Inequality Signs < means“is less than”  means “is less than or equal to” > means “is greater than”  means ”is greater than or equal to” Reading Inequalities 1. 3 < n n is greater than 3. 2. 4  n n is greater than or equal to 4. 3. -2  n  1 n is greater than or equal to -2 but less than or equal to 1. 6 < 8 or 8 > 6
• 4.
  Read the followinginequalities 1. 0 < a < 3 a is greater than 0 but less than 3. 2. -5  b < 2 b is greater than or equal to -5 but less than 2. 3. -7  d  -1 d is greater than or equal to -7 but less than or equal to -1. 4. 9 > e e is less than 9 5. f > -1 f is greater than -1 6. 6  g g is less than or equal to 6 7. h  - 4 h is greater than or equal to - 4
• 5.
  Displaying Inequalities ona Number Line 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 j  1 k  -2 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 m > 4 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 n > 0
• 6.
  0 1 23 4 5 6 -1 -2 -3 -4 -5 -6 p  1 q  -2 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 r < 4 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 s < 0 Displaying Inequalities on a Number Line
• 7.
  0 1 23 4 5 6 -1 -2 -3 -4 -5 -6 -5  u  5 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -1 < v  4 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -5 < w < 0 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -3  x < 6 Displaying Inequalities on a Number Line
• 8.
  0 1 23 4 5 6 -1 -2 -3 -4 -5 -6 -5  y  5 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -1 < z  4 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -5 < a < 0 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 -3  b < 6 Displaying Inequalities on a Number Line
• 9.
  State the inequalitiesdisplayed on each number line below. 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 0 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 1 2 3 4 0  x  5 -5 < y  4 0 < n < 5 -5  m < - 3
• 10.
  3x + 2> 5x – 12 3x + 2 = 5x – 12 What is the difference between the two? 2 > 2x - 12 14 > 2x 7 > x Or x < 7 2 = 2x - 12 14 = 2x 7 = x Or x = 7
• 11.
  Example Find the rangeof values of x that satisfy both 3x ≥ 2(x – 1) and 10 - 3x > 6. 3x ≥ 2(x – 1) 10 - 3x > 6 3x ≥ 2x – 2 x ≥ -2 10 > 6 + 3x 4 > 3x > x OR x < 4 3 4 3 And so -2 ≤ x < 4 3
• 12.
  Inequalities and Regions x y 01 2 3 4 5 -1 -2 -3 -4 1 2 3 4 5 -1 -2 -3 -4 -5 Shade the region for which x + 2y ≥ 6 1. Draw the boundary line equation x + 2y = 6. 2. Choose a test point in one of the 2 regions to see whether or not it satisfies the inequality, then shade the required region. (2,4) 2 + 2 x 4 = 10 ≥ 6  Boundary line solid if inequality is either ≤ or ≥ x + 2y = 6  y = -½x + 3 Finding the region for a single inequality y intercept 3, gradient –½
• 13.
  x y 0 1 23 4 5 -1 -2 -3 -4 1 2 3 4 5 -1 -2 -3 -4 -5 Shade the region for which 2x - y < -1 1. Draw the boundary line equation 2x - y = -1 2. Choose a test point in one of the 2 regions to see whether or not it satisfies the inequality then shade the required region. (3,1) 2 x 3 - 1 = 5 > -1  2x - y = -1  y = 2x + 1 Boundary line dotted if inequality is either < or > Inequalities and Regions Finding the region for a single Inequality y intercept 1, gradient 2
• 14.
  Inequalities and Regions Findingthe region for two inequalities x y 0 1 2 3 4 5 -1 -2 -3 -4 1 2 3 4 5 -1 -2 -3 -4 -5 Shade and label with the letter R, the region for which y ≥ 1 and x > 2. Draw boundary line y = 1 Note or lightly shade the region for which y ≥ 1. Draw boundary line x = 2 Note or lightly shade the region for which x > 2 R Identify the overlapping region that satisfies both inequalities and label. Boundary line solid since inequality is ≥ Boundary line dotted since inequality is >
• 15.
  x y 0 1 23 4 5 -1 -2 -3 -4 1 2 3 4 5 -1 -2 -3 -4 -5 Shade and label with the letter R, the region for which x + y < -2 and x ≤ 1 Draw line x + y = -2  y = -x – 2 Note or lightly shade the region for which x + y < -2 Draw line x = 1 Note or lightly shade the region for which x ≤ 1 Identify the overlapping region that satisfies both inequalities and label. Boundary line dotted since inequality is < Boundary line solid since inequality is ≤ R The origin (0.0) makes a useful test point. y intercept -2 and gradient -1
• 16.
  Inequalities and Regions Inequalitiesthat enclose a region of the plane. x y 0 1 2 3 4 5 -1 -2 -3 -4 1 2 3 4 5 -1 -2 -3 -4 -5 Shade and label with the letter R, the region for which y ≥ -3 and x > 1 and 2x + y < 3 Draw line y = -3 Note or lightly shade the region for which y ≥ -3. Draw line x = 1 Note or lightly shade the region for which x > 1 Draw line 2x + y = 3  y = -2x + 3 y intercept 3 and gradient - 2 Note or lightly shade the region for which 2x + y < 3 The origin (0.0) makes a useful test point. R Identify the overlapping region that satisfies all 3 inequalities and label.
• 17.
  x y 0 1 23 4 5 -1 -2 -3 -4 1 2 3 4 5 -1 -2 -3 -4 -5 Shade and label with the letter R, the region for which y ≥ -3, x > -2, y  2x - 3 and x + y < 2 Draw line y = -3 Note or lightly shade the region for which y ≥ -3. Draw line x = -2 Note or lightly shade the region for which x > -2 Draw line y = 2x – 3 Note or lightly shade the region for which y  2x - 3 The origin (0.0) makes a useful test point. y intercept -3 and gradient 2. The origin (0.0) makes a useful test point. R Draw line x + y = 2  y = -x + 2 y intercept 2, gradient 1 Note or lightly shade the region x + y < 2 Identify the overlapping region that satisfies all 4 inequalities and label.