6.1: Inequalities Using Addition and Subtraction
Learning Objectives
 Write and graph inequalities in one variable on a number line.
 Solve an inequality using addition.
 Solve an inequality using subtraction.
Introduction
Inequalities are similar to equations in that they show a relationship between two expressions. We solve and graph inequalities in a similar way to equations. However, there are some differences that we will talk about in this chapter. The main difference is that for linear inequalities the answer is an interval of values whereas for a linear equation the answer is most often just one value.
When writing inequalities we use the following symbols
Write and Graph Inequalities in One Variable on a Number Line
Let’s start with the simple inequality
We read this inequality as “
Consider another simple inequality
We read this inequality as “
In a graph, we use an empty circle for the endpoint of a strict inequality
Example 1
Graph the following inequalities on the number line.
a)
b)
c)
d)
Solution
a) The inequality
b) The inequality
c) The inequality
d) The inequality
Example 2
Write the inequality that is represented by each graph.
a)
b)
c)
d)
Solution:
a)
b)
c)
d)
Inequalities appear everywhere in real life. Here are some simple examples of realworld applications.
Example 3
Write each statement as an inequality and graph it on the number line.
a) You must maintain a balance of at least $2500 in your checking account to get free checking.
b) You must be at least 48 inches tall to ride the “Thunderbolt” Rollercoaster.
c) You must be younger than 3 years old to get free admission at the San Diego Zoo.
d) The speed limit on the interstate is 65 miles per hour.
Solution:
a) The inequality is written as
b) The inequality is written as
c) The inequality is written as
d) Speed limit means the highest allowable speed, so the inequality is written as
Solve an Inequality Using Addition
To solve an inequality we must isolate the variable on one side of the inequality sign. To isolate the variable, we use the same basic techniques used in solving equations. For inequalities of this type:
We isolate the
Example 4
Solve each inequality and graph the solution set.
a)
b)
c)
d)
Solution:
a)
b)
c)
d)
Solve an Inequality Using Subtraction
For inequalities of this type:
We isolate the
Example 5
Solve each inequality and graph the solution set.
a)
b)
c)
d)
Solution:
a)
b)
c)
d)
Lesson Summary
 The answer to an inequality is often an interval of values. Common inequalities are:
 > is greater than

≥ is greater than or equal to  > is less than

≤ is less than or equal to  Solving inequalities with addition and subtraction works just like solving an equation. To solve, we isolate the variable on one side of the equation.
Review Questions
 Write the inequality represented by the graph.
 Write the inequality represented by the graph.
 Write the inequality represented by the graph.
 Write the inequality represented by the graph.
Graph each inequality on the number line.

x<−35  \begin{align*} x > 17\end{align*}
 \begin{align*} x \geq 20\end{align*}
 \begin{align*} x \leq 3 \end{align*}
Solve each inequality and graph the solution on the number line.
 \begin{align*} x5 < 35\end{align*}
 \begin{align*} x+ 15 \geq 60\end{align*}
 \begin{align*} x2 \leq 1\end{align*}
 \begin{align*} x8 > 20\end{align*}
 \begin{align*} x+11 > 13\end{align*}
 \begin{align*} x+ 65 < 100 \end{align*}
 \begin{align*} x32 \leq 0\end{align*}
 \begin{align*} x+68 \geq 75\end{align*}
Review Answers
 \begin{align*} x \geq 1\end{align*}
 \begin{align*} x <10 \end{align*}
 \begin{align*} x \leq 10\end{align*}
 \begin{align*}x > 30\end{align*}
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