2.9.5 Comparing Equality and Inequality

Comparing Equality and Inequality

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

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

Example 1

$2x-3<5$

We can solve its related equation.

$\begin{array}{ccc}\hfill 2x-3& \hfill =\hfill & 5\hfill \\ \hfill 2x& \hfill =\hfill & 8\hfill \\ \hfill x& \hfill =\hfill & 4\hfill \end{array}$

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?

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

$5\stackrel{?}{<}5$

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

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 $x=8$.

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

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

So, 8 does not satisfy the inequality since $13\nless 5$. 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 $x=0$.

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

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

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.

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

We can solve its related equation.

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

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

Does y = 3 satisfy the inequality?

$3(3)-7\stackrel{?}{\ge }8-2(3)$

$2\stackrel{?}{\ge }2$

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 $y=5$.

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

$1\stackrel{?}{\ge }-2$

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.

$3(0)-7\stackrel{?}{\ge }8-2(0)$

$-7\stackrel{?}{\ge }8$

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 $y\ge 3$.

Example 3

Look at the inequality

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

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

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

Does 3 satisfy the inequality?

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

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

$-4$

$\text{⧸}<$

$-4$

So, 3 is not a solution.

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

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

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

$-\frac{11}{2}$

$\text{⧸}<$

$-10$

So, 5 is not a solution

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

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

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

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

So, 0 is a solution.

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