10.6 solving linear inequalities

Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
Equations,
Inequalities, and
Applications
10

1. Graph intervals on a number line.
2. Use the addition property of inequality.
3. Use the multiplication property of inequality.
4. Solve inequalities using both properties of
inequality.
5. Use inequalities to solve applied problems.
6. Solve linear inequalities with three parts.
Objectives
10.6 Solving Linear Inequalities

Inequalities are algebraic expressions related by
< “is less than,” ≤ “is less than or equal to,”
> “is greater than,” ≥ “is greater than or equal to.”
We solve an inequality by finding all real number
solutions for it. For example, the solutions of x ≤ 2
include all real numbers that are less than or equal to 2,
not just the integers less than or equal to 2.

Linear Inequality in One Variable
A linear inequality in one variable can be written in
the form
Ax + B < C,
where A, B, and C are real numbers, with A ≠ 0.
Examples of linear inequalities in one variable include
x + 5 < 2, t – 3 ≥ 5, and 2k + 5 ≤ 10.
,Ax B C  ,Ax B C  ,Ax B C 

Graphing an Interval on a Number Line
Example Graph x > –2.
The statement x > –2 says that x can represent any value
greater than –2 but cannot equal –2 itself. We show this
interval on a graph by placing a parenthesis at –2 and drawing
an arrow to the right. The parenthesis at –2 indicates that –2 is
not part of the graph.
(

Graphing an Interval on a Number Line
Example Graph 3 > x.
The 3 > x means the same as x < 3. The inequality symbol
continues to point to the lesser value. The graph of x < 3 in
interval notation is written as  ,3 .
)

Addition Property of Inequality
For any real numbers A, B, and C, the inequalities
A < B and A + C < B + C
have exactly the same solutions.
In words, the same number may be added to each
side of an inequality without changing the solutions.
Use the Addition Property of Inequality
Note
As with the addition property of equality, the same
number may be subtracted from each side of an inequality.

Example Solve 5 + 6x ≤ 5x + 8, and graph the solution set.
5 + 6x ≤ 5x + 8
5 + x ≤ 8
x ≤ 3
Use the Addition Property of Inequality
–5x –5x
–5 –5
A graph of the solution set is
]

Multiplication Property of Inequality
For any real numbers A, B, and C (C ≠ 0),
1. If C is positive, then the inequalities
A < B and AC < BC
are equivalent.*
2. If C is negative, then the inequalities
A < B and AC > BC
are equivalent.*
In words, each side of an inequality may be multi-
plied by the same positive number without changing
the solutions. If the multiplier is negative, we must
reverse the direction of the inequality symbol.
*This also applies to A ≤ B, A > B, and A ≥ B.
Use the Multiplication Property of Inequality

Use the Multiplication Property of Inequality
Note
As with the multiplication property of equality, the
same nonzero number may be divided into each side.
If the divisor is negative, we must reverse the
direction of the inequality.
This property of inequality holds for any type of
inequality (<, >, ≤, and ≥).

Example Solve 6y > 12, and graph the solution set.
6y > 12
y > 2
Using the Mulitiplication Property of Inequality
6 6
(

Solving a Linear Inequality
Step 1 Simplify each side separately. Use the
distributive property to clear parentheses, clear any
fractions or decimals and combine terms.
Step 2 Isolate the variable term on one side. Use the
addition property of inequality so that all terms with
variables are on one side of the inequality and all
constants (numbers) are on the other side.
Step 3 Isolate the variable. Use the multiplication
property of inequality to obtain an inequality in one
of the following forms, where k is a constant
(number) variable < k, variable ≤ k , variable > k
or variable ≥ k .
Solve Inequalities Using Both Properties of Inequality

Reverse the direction of the
inequality symbol when
dividing each side by a
negative number.
Example Solve –2(z + 3) – 5z ≤ 4(z – 1) + 9.
Graph the solution set.
–2(z + 3) – 5z ≤ 4(z – 1) + 9
–11z – 6 ≤ 5
–11z ≤ 11
–4z –4z
+6 + 6
Solving a Linear Inequality
–2z – 6 – 5z ≤ 4z – 4 + 9
–7z – 6 ≤ 4z + 5
11 11
z ≥ –1
[

Phrase Example Inequality
Is greater than A number is greater than 4 x > 4
Is less than A number is less than –12 x < –12
Is at least A number is at least 6 x ≥ 6
Is at most A number is at most 8 x ≤ 8
Use Inequalities to Solve Applied Problems

Brent has test grades of 86, 88,
and 78 on his first three tests in
geometry. If he wants an
average of at least 80 after his
fourth test, what are the possible
scores he can make on that test?
Let x = Brent’s score on his
fourth test. To find his average
after four tests, add the test
scores and divide by 4.
86 88 78
80
4
x  

4 4
 
 
 
Example
Use Inequalities to Solve Applied Problems
252 + x ≥ 320
252
80
4
x

–252 –252
x ≥ 68
Brent must score 68 or more
on the fourth test to have an
average of at least 80.

Solve Linear Inequalities With Three Parts
Example Graph –1 < x ≤ 2.
The statement –1 < x ≤ 2 is read “–1 is less than x and x is less
than or equal to 2.” We graph the solutions to this inequality
by placing a parenthesis at –1 (because –1 is not part of the
graph) and a bracket at 2 (because 2 is part of the graph), then
drawing a line segment between the two. Notice that the graph
includes all points between –1 and 2 and includes 2 as well.
( ]

Solve Linear Inequalities With Three Parts
Example Solve 4 3 5 10  x
4 3 55 5 10 5    x
9 3 15 x
3 3
3 1
3
9 5
 
x
3 5 x
[ )

10.6 solving linear inequalities

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