Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

2.9.5 Comparing Equality and Inequality

2.9.5 • Comparing Equality and Inequality

Activity

Use this equation for questions 1 – 2:

$- \frac{4 (x + 3)}{5} = 4 x - 12$

1.

Solve this equation. What is the value of $x$ ?

2

2.

Check your solution.

Compare your answer:

$\begin{matrix} - \frac{4 (x + 3)}{5} & = 4 x - 12 \\ - \frac{4 (2 + 3)}{5} & = 4 (2) - 12 \\ - \frac{4 (5)}{5} & = 8 - 12 \\ - \frac{20}{5} & = - 4 \\ - 4 & = - 4 ✓ \end{matrix}$

Use this inequality for questions 3 and 6:

$- \frac{4 (x + 3)}{5} \leq 4 x - 12$

3.

Choose a couple of values less than 2 for $x$ . Are they solutions to the inequality?

Compare your answer: Values less than 2 are not solutions.

4.

Choose a couple of values greater than 2 for $x$ . Are they solutions to the inequality?

Compare your answer: Values greater than 2 are solutions.

5.

Choose 2 for $x$ . Is it a solution to the inequality?

Compare your answer: Yes, 2 is a solution.

6.

Graph the solution to the inequality on the number line.

Compare your answer:

Are you ready for more?

Extending Your Thinking

Here is a different type of inequality: $x^{2} \leq 4$ .

1.

Which of the following values solve the inequality: 1, 3 or -3?

Compare your answer:

From the given values, only 1 is a solution of the inequality.

2.

Describe all solutions to this inequality. (If you like, you can graph the solutions on a number line.)

Compare your answer:

$- 2 \leq x \leq 2$

3.

Describe all solutions to the inequality $x^{2} \geq 9$ . Test several numbers to make sure your answer is correct.

Compare your answer:

$x \leq - 3$ or $x \geq 3$

Self Check

Which number line represents the solution to the inequality?

\frac12(–2z\;+\;2)\;\geq\;12

Additional Resources

Comparing Equality and Inequality

We can use related equations to help find solutions to inequalities.

Example 1

Here’s an inequality with one variable on one side of the equation.

$2 x - 3 < 5$

We can solve its related equation.

$\begin{matrix} 2 x - 3 & = & 5 \\ 2 x & = & 8 \\ x & = & 4 \end{matrix}$

Now, let’s verify which solutions verify the original inequality.

Step 1 - Verify if the solution to the related equation satisfies the original inequality.

Does $x = 4$ satisfy the inequality?

$2 (4) - 3 \overset{?}{<} 5$

$5 \overset{?}{<} 5$

No, 4 is not a solution since it the $5 ≮ 5$ .

Step 2 - Verify if a value greater than the solution to the related equation satisfies the original inequality.

Let’s try a number bigger than 4. For example, try the value of $x = 8$ .

$2 (8) - 3 \overset{?}{<} 5$

$13 \overset{?}{<} 5$

So, 8 does not satisfy the inequality since $13 ≮ 5$ . That means $x$ is not a value greater than 4.

Step 3 - Verify if a value less than the solution to the related equation satisfies the original inequality.

Let’s try a number smaller than 4, such as $x = 0$ .

$2 (0) - 3 \overset{?}{<} 5$

$- 3 \overset{?}{<} 5$

Thus, 0 satisfies the inequality since $- 3 < 5$ . Since 0 is a number less than 4, this means the solution set is $x < 4$ .

Example 2

Let’s solve an inequality with the same variable on both sides of the equation.

$3 y - 7 \geq 8 - 2 y$

We can solve its related equation.

$\begin{matrix} 3 y - 7 & = & 8 - 2 y \\ 5 y & = & 15 \\ y & = & 3 \end{matrix}$

Step 1 - Verify if the solution to the related equation satisfies the original inequality.

Does $y$ = 3 satisfy the inequality?

$3 (3) - 7 \overset{?}{\geq} 8 - 2 (3)$

$2 \overset{?}{\geq} 2$

Yes, $y = 3$ satisfies the inequality because 2 is greater than or equal to 2.

Step 2 - Verify if a value greater than the solution to the related equation satisfies the original inequality.

Let’s try a number greater than 3 such as $y = 5$ .

$3 (5) - 78 - 2 (5)$

$1 \overset{?}{\geq} - 2$

So, 5 satisfies the inequality since the value of 1 is greater than or equal to the value of -2.

Step 3 - Verify if a value less than the solution to the related equation satisfies the original inequality.

Let’s try a number less than $y = 3$ such as 0.

$3 (0) - 7 \overset{?}{\geq} 8 - 2 (0)$

$- 7 \overset{?}{\geq} 8$

So, $y = 0$ does not satisfy the inequality. This means that values less than $y = 3$ are not part of the solution set.

We verified that $y = 3$ and values greater than 3 satisfy the inequality $(y > 3)$ , so the solution set is $y \geq 3$ .

Example 3

Look at this more complex inequality:

$- \frac{3 x + 7}{4} < - 3 x + 5$

We can solve its related equation $- \frac{3 x + 7}{4} = - 3 x + 5$ .

After solving, we know that $x = 3$ .

Does 3 satisfy the inequality?

$- \frac{3 x + 7}{4} < - 3 x + 5$

$- \frac{3 (3) + 7}{4} \overset{?}{<} - 3 (3) + 5$

$- 4 ⧸ < - 4$

So, 3 is not a solution.

Let’s try a number bigger than 3, such as 5.

$- \frac{3 x + 7}{4} < - 3 x + 5$

$- \frac{3 (5) + 7}{4} \overset{?}{<} - 3 (5) + 5$

$- \frac{11}{2} ⧸ < - 10$

So, 5 is not a solution

Let’s try a number smaller than 3, such as 0.

$- \frac{3 x + 7}{4} < - 3 x + 5$

$- \frac{3 (0) + 7}{4} \overset{?}{<} - 3 (0) + 5$

$- \frac{7}{4} < 5$

So, 0 is a solution.

This means that the solution set is $x < 3$ .

Try it

Comparing Equality and Inequality

Use the related equation to find the solution set to the inequality.

$\frac{5 x - 3}{2} < 3 x - 4$

1.

What is the related equation?

Compare your answer: $\frac{5 x - 3}{2} = 3 x - 4$

2.

What is the solution to the related equation?

Compare your answer:

$x = 5$

3.

What values did you test on the inequality?

Compare your answer: Your answer may vary, but here is a sample. Since 5 is the solution to the related equation, I tested it. I also tested 4 and 6 since they represent a value less than and greater than 5.

4.

What is the solution set to the inequality?

Compare your answer: $x > 5$