Download Solving Linear Inequalities in One Variable

Equations and Inequalities
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2
Section
2.7
Solving Linear Inequalities in
One Variable
Copyright © Cengage Learning. All rights reserved.
Objectives
1. Solve a linear inequality in one variable using
1
the properties of inequality and graph the
solution on a number line.
2. Solve a compound linear inequality in one
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variable.
3. Solve an application involving a linear
3
inequality in one variable.
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1. Solve a linear inequality in one variable
using the properties of inequality and graph
the solution on a number line
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Solving an Inequality
Recall the meaning of the following symbols.
Inequality Symbols
 means “is less than”
 means “is greater than”
 means “is less than or equal to”
 means “is greater than or equal to”
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Solving an Inequality
A solution of an inequality is any number that makes the
inequality true.
The number 2 is a solution of the inequality
x3
because 2  3.
This inequality has many more solutions, because any real
number that is less than or equal to 3 will satisfy it.
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Solving an Inequality
Using a graph to solve inequality: x  3
The red arrow in Figure 2-16 indicates all those points with
coordinates that satisfy the inequality x  3.
Figure 2-16
The bracket at the point with coordinate 3 indicates that the
number 3 is a solution of the inequality x  3.
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Solving an Inequality
The graph of the inequality x  1 appears in Figure 2-17.
Figure 2-17
The red arrow indicates all those points whose coordinates
satisfy the inequality.
The parenthesis at the point with coordinate 1 indicates
that 1 is not a solution of the inequality x  1.
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Solving an Inequality
Addition Property and Subtraction Property of Inequality
Suppose a, b, and c are real numbers.
•If a  b, then a + c  b + c.
•If a  b, then a – c  b – c.
Similar statements can be made for the symbols , , and .
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Solving an Inequality
The addition property of inequality can be stated this
way:
If any quantity is added to both sides of an inequality, the
resulting inequality has the same direction as the original
Inequality.
The subtraction property of inequality can be stated this
way:
If any quantity is subtracted from both sides of an
inequality, the resulting inequality has the same direction
as the original inequality.
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Example
Solve 2x + 5  x – 4 and graph the solution on a number
line.
Solution:
To isolate the x on the left side of the  sign, we proceed as
if we were solving an equation.
2x + 5  x – 4
2x + 5 – 5  x – 4 – 5
2x  x – 9
2x – x  x – 9 – x
x  –9
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Example – Solution
cont’d
The graph of the solution (see Figure 2-18) includes all
points to the right of –9 but does not include –9 itself. For
this reason, we use a parenthesis at –9.
Figure 2-18
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Solving an Inequality
If both sides of the true inequality 6  9 are multiplied or
divided by a positive number, such as 3, another true
inequality results.
69
3639
18  27
69
Multiply both sides by 3.
True.
Divide both sides by 3.
23
True.
The inequalities 18  27 and 2  3 are true.
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Solving an Inequality
However, if both sides of 6  9 are multiplied or divided by
a negative number, such as –3, the direction of the
inequality symbol must be reversed to produce another true
inequality.
69
–3  6  –3  9
–18  –27
69
Multiply both sides
by –3 and reverse
the direction of the
inequality.
True.
Divide both sides
by –3 and reverse
the direction of the
inequality.
–2  –3
True.
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Solving an Inequality
The inequality –18  –27 is true, because –18 lies to the
right of –27 on the number line.
The inequality –2  –3 is true, because –2 lies to the right
of –3 on the number line.
This example suggests the multiplication and division
properties of inequality.
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Solving an Inequality
Multiplication Property of Inequality
Suppose a, b, and c are real numbers.
If a  b and c  0, then ac  bc.
If a  b and c  0, then ac  bc.
Division Property of Inequality
Suppose a, b, and c are real numbers.
If a  b and c  0, then
.
If a  b and c  0, then
.
Similar statements can be made for the symbols , , and .
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Solving an Inequality
The multiplication property of inequality can be stated
this way:
If unequal quantities are multiplied by the same positive
quantity, the results will be unequal and in the same
direction as the original inequality.
If unequal quantities are multiplied by the same negative
quantity, the results will be unequal but in the opposite
direction of the original inequality.
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Solving an Inequality
The division property of inequality can be stated this
way:
If unequal quantities are divided by the same positive
quantity, the results will be unequal and in the same
direction as the original inequality.
If unequal quantities are divided by the same negative
quantity, the results will be unequal but in the opposite
direction of the original inequality.
To divide both sides of an inequality by a nonzero
number c, we could instead multiply both sides by .
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Solving an Inequality
Note that the procedures for solving inequalities are the
same as for solving equations, except that we must reverse
the inequality symbol whenever we multiply or divide by a
negative number.
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2.
Solve a compound linear
inequality in one variable
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Solving a Compound Inequality
Two inequalities often can be combined into a double
inequality or compound inequality to indicate that
numbers lie between two fixed values.
For example, the inequality 2  x  5 indicates that x is
greater than 2 and that x is also less than 5. The solution of
2  x  5 consists of all numbers that lie between 2 and 5.
The graph of this set (called an interval) appears in
Figure 2-21.
Figure 2-21
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Example
Solve –4  2(x – 1)  4, and graph the solution on the
number line.
Solution:
To isolate x in the center, we proceed as if we were solving
an equation with three parts: a left side, a center, and a
right side.
–4  2(x – 1)  4
–4  2x – 2  4
Use the distributive property to
remove parentheses.
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Example – Solution
–2  2x  6
cont’d
Add 2 to all three parts.
–1  x  3
Divide all three parts by 2.
The graph of the solution appears in Figure 2-22.
Figure 2-22
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3. Solve an application involving a linear
inequality in one variable
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Solving an Inequality
When solving applications, there are certain words that
help us translate a sentence into a mathematical inequality.
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Example – Grades
A student has scores of 72, 74, and 78 points on three
mathematics examinations. How many points does he
need on his last exam to earn a B or better, an average of
at least 80 points?
Solution:
We can let x represent the score on the fourth (last) exam.
To find the average grade, we add the four scores and
divide by 4.
To earn a B, this average must be greater than or equal to
80 points.
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Example – Solution
cont’d
We can solve this inequality for x.
Add.
224 + x  320
x  96
Multiply both sides by 4.
Subtract 224 from both sides.
To earn a B, the student must score at least 96 points.
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Your Turn
Solve the following inequalities for x and graph the solution:
1. 2x + 5 < 13
2. -4 < 3x + 5 < 13
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