6.3.3: Efficiently Solving Inequalities

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Illustrative Mathematics
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Lesson

Let's solve more complicated inequalities.

Exercise $\PageIndex{1}$: Lots of Negatives

Here is an inequality: $-x\geq -4$.

Predict what you think the solutions on the number line will look like.
Select all the values that are solutions to $-x\geq -4$:
1. $3$
2. $-3$
3. $4$
4. $-4$
5. $4.001$
6. $-4.001$
Graph the solutions to the inequality on the number line:

Exercise $\PageIndex{2}$: Inequalities with Tables

1. Let's investigate the inequality $x-3>-2$.

Table $\PageIndex{1}$
$x$	$-4$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$x-3$	$-7$		$-5$				$-1$		$1$

Complete the table.
For which values of $x$ is it true that $x-3=-2$?
For which values of $x$ is it true that $x-3>-2$?
Graph the solutions to $x-3>-2$ on the number line:

2. Here is an inequality: $2x<6$.

Predict which values of $x$ will make the inequality $2x<6$ true.

Complete the table. Does it match your prediction?

Table $\PageIndex{2}$
$x$	$-4$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$2x$

Graph the solutions to $2x<6$ on the number line:

3. Here is an inequality: $-2x<6$.

Predict which values of $x$ will make the inequality $-2x<6$ true.

Complete the table. Does it match your prediction?

Table $\PageIndex{4}$
$x$	$-4$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$-2x$

Graph the solutions to $-2x<6$ on the number line:

d. How are the solutions to $2x<6$ different from the solutions to $-2x<6$?

Exercise $\PageIndex{3}$: Which Side are the Solutions?

1. Let's investigate $-4x+5\geq 25$.

Solve $-4x+5=25$.
Is $-4x+5\geq 25$ true when $x$ is 0? What about when $x$ is $7$? What about when $x$ is $-7$?
Graph the solutions to $-4x+5\geq 25$ on the number line.

2. Let's investigate $\frac{4}{3}x+3<\frac{23}{3}$.

Solve $\frac{4}{3}x+3=\frac{23}{3}$.
Is $\frac{4}{3}x+3<\frac{23}{3}$ true when $x$ is $0$?
Graph the solutions to $\frac{4}{3}x+3<\frac{23}{3}$ on the number line.

3. Solve the inequality $3(x+4)>17.4$ and graph the solutions on the number line.

4. Solve the inequality $-3\left(x-\frac{4}{3}\right)\leq 6$ and graph the solutions on the number line.

Are you ready for more?

Write at least three different inequalities whose solution is $x>-10$. Find one with $x$ on the left side that uses a $<$.

Summary

Here is an inequality: $3(10-2x)<18$. The solution to this inequality is all the values you could use in place of $x$ to make the inequality true.

In order to solve this, we can first solve the related equation $3(10-2x)=18$ to get the solution $x=2$. That means 2 is the boundary between values of that make the inequality true and values that make the inequality false.

To solve the inequality, we can check numbers greater than 2 and less than 2 and see which ones make the inequality true.

Let’s check a number that is greater than 2: $x=5$. Replacing $x$ with 5 in the inequality, we get $3(10-2\cdot 5)<18$ or just $0<18$. This is true, so $x=5$ is a solution. This means that all values greater than 2 make the inequality true. We can write the solutions as $x>2$ and also represent the solutions on a number line:

Notice that 2 itself is not a solution because it's the value of $x$ that makes $3(10-2x)$ equal to 18, and so it does not make $3(10-2x)<18$ true.

For confirmation that we found the correct solution, we can also test a value that is less than 2. If we test $x=0$, we get $3(10-2\cdot 0)<18$ or just $30<18$. This is false, so $x=0$ and all values of $x$ that are less than 2 are not solutions.

Glossary Entries

Definition: Solution to an Inequality

A solution to an inequality is a number that can be used in place of the variable to make the inequality true.

For example, 5 is a solution to the inequality $c<10$, because it is true that $5<10$. Some other solutions to this inequality are 9.9, 0, and -4.

Practice

Exercise $\PageIndex{4}$

Consider the inequality $-1\leq \frac{x}{2}$.

Predict which values of $x$ will make the inequality true.

Complete the table to check your prediction.

Table $\PageIndex{5}$
$x$	$-4$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$\frac{x}{2}$

Consider the inequality $1\leq\frac{-x}{2}$.

Predict which values of $x$ will make it true.

Complete the table to check your prediction.

Table $\PageIndex{6}$
$x$	$-4$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$-\frac{x}{2}$

Exercise $\PageIndex{5}$

Diego is solving the inequality $100-3x\geq -50$. He solves the equation $100-3x=-50$ and gets $x=50$. What is the solution to the inequality?

$x<50$
$x\leq 50$
$x>50$
$x\geq 50$

Exercise $\PageIndex{6}$

Solve the inequality $-5(x-1)>-40$, and graph the solution on a number line.

Exercise $\PageIndex{7}$

Select all values of $x$ that make the inequality $-x+6\geq 10$ true.

$-3.9$
$4$
$-4.01$
$-4$
$4.01$
$3.9$
$0$
$-7$

(From Unit 6.3.1)

Exercise $\PageIndex{8}$

Draw the solution set for each of the following inequalities.

1. $x>7$

2. $x\geq -4.2$

(From Unit 6.3.1)

Exercise $\PageIndex{9}$

The price of a pair of earrings is $22 but Priya buys them on sale for $13.20.

By how much was the price discounted?
What was the percentage of the discount?

(From Unit 4.3.3)

\(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(x-3\)	\(-7\)		\(-5\)				\(-1\)		\(1\)

\(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(2x\)

\(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(-2x\)

\(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(\frac{x}{2}\)

\(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(-\frac{x}{2}\)

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