Inequality problems and exercises can be solved with a process similar to the one we use to solve equations. The main difference regarding inequalities is that we have to change the side of the inequality sign when we multiply or divide by negative numbers.

Here, we will look at a summary of how to solve inequalities. In addition, we will look at several examples with answers to master the process of solving inequalities.

## Process used to solve inequalities

Remember that inequalities are relationships that compare two values using the signs greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). For example, $latex 3x<6$ and $latex 2x+2>3$ are inequalities.

To solve inequalities, we can follow the following steps:

**Step 1:** We simplify the inequality if possible. This includes removing grouping signs such as parentheses, combining like terms, and removing fractions.

** Step 2:** Solve for the variable. We have to do addition and subtraction so that all the variables are located on one side of the inequality and the constants are located on the other side.

** Step 3:** Solve. We use division or multiplication to find the answer. Note: When we multiply or divide an inequality by a negative number, we must switch sides to the inequality sign.

** Step 4:** If we have to graph, we have to remember that we use an empty point to indicate that the limiting number is not part of the solution and we use a filled point to indicate that the limiting number is part of the solution.

For example, if the solution is $latex x>2$, the 2 is not part of the solution, so we use an empty point and if the solution is $latex x \ge 2$, the 2 is part of the solution, so we use a filled point.

## Inequalities – Examples with answers

The following examples of inequalities with answers help us to fully master solving inequalities. Each example has a detailed solution that indicates the process to follow to find the solution.

**EXAMPLE 1**

Solve and graph the inequality $latex 5x-10<15$.

##### Solution

**Step 1:** Here, we have nothing to simplify, so we start with:

$latex 5x-10<15$

**Step 2:** To solve for the variable, we add 10 from both sides and simplify:

$latex 5x-10+10<15+10$

$latex 5x<25$

**Step 3:** To solve, we divide both sides by 5:

$latex \frac{5}{5}x<\frac{25}{5}$

$latex x<5$

**Step 4:** To graph, we note that the solutions to the inequality are all real numbers to the left of 5. The 5 is not included, so we use an empty point to indicate this:

**EXAMPLE 2**

Solve and graph the inequality $latex -4x-5\leq 3$.

##### Solution

**Step 1:** We have nothing to simplify, so we start with:

$latex -4x-5\leq 3$

**Step 2:** We add 5 to both sides to solve for the variable:

$latex -4x-5+5\leq 3+5$

$latex -4x\leq 8$

**Step 3:** We divide both sides by -4 to get:

$latex \frac{-4}{-4}x\leq\frac{8}{-4}$

$latex x\geq -2$

Don’t forget to change the inequality sign when multiplying or dividing by a negative number. |

**Step 4:** In this case, -2 is part of the solution. Therefore, we use a solid point to indicate that the solutions are all numbers to the right of -2, including -2:

**EXAMPLE 3**

Solve the inequality $latex 5x+3>3x-3$.

##### Solution

**Step 1:** We have nothing to simplify. We start with the inequality:

$latex 5x+3>3x-3$

**Step 2:** We subtract 3 and 3x from both sides to solve for the variable:

$latex 5x+3-3-3x>3x-3-3-3x$

$latex 2x>-6$

**Step 3:** We divide both sides by 2 to solve:

$latex \frac{2}{2}x>\frac{-6}{2}$

$latex x>-3$

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**EXAMPLE 4**

Solve the inequality $latex 3(x+2)>-9$.

##### Solution

**Step 1:** We have parentheses, so we apply the distributive property to eliminate them:

$latex 3(x+2)>-9$

$latex 3x+6>-9$

**Step 2:** To solve for the variable, we subtract 6 from both sides:

$latex 3x+6-6>-9-6$

$latex 3x>-15$

**Step 3:** To solve, we divide both sides by 3:

$latex \frac{3}{3}x>\frac{-15}{3}$

$latex x>-5$

**EXAMPLE 5**

Solve the inequality $latex 2(2x+4)+5>1$.

##### Solution

**Step 1:** We simplify the parentheses and combine like terms:

$latex 2(2x+4)+5>1$

$latex 4x+8+5>1$

$latex 4x+13>1$

**Step 2:** We isolate the variable by subtracting 13 from both sides:

$latex 4x+13-13>1-13$

$latex 4x>-12$

**Step 3:** We have to divide by 4:

$latex \frac{4}{4}x>\frac{-12}{4}$

$latex x>-3$

**EXAMPLE 6**

Solve the inequality $latex 4(2x+4)-3\leq 2(3x+4)+3$.

##### Solution

**Step 1:** We simplify the parentheses on both sides and combine like terms:

$latex 4(2x+4)-3\leq 2(3x+4)+3$

$latex 8x+16-3\leq 6x+8+3$

$latex 8x+13\leq 6x+11$

**Step 2:** We subtract 13 and 6x from both sides to solve for the variable:

$$8x+13-13-6x\leq 6x+11-13-6x$$

$latex 2x\leq -2$

**Step 3:** To solve, we divide both sides by 2:

$latex \frac{2}{2}x\leq \frac{-2}{2}$

$latex x\leq -2$

**EXAMPLE 7**

Solve the inequality $latex 2(x+5)-10\geq 4(2x+6)$.

##### Solution

**Step 1:** We simplify the parentheses on both sides and combine like terms:

$latex 2(x+5)-10\geq 4(2x+6)$

$latex 2x+10-10\geq 8x+24$

$latex 2x\geq 8x+24$

**Step 2:** We subtract both sides by 8x to solve for *x*:

$latex 2x-8x\geq 8x+24-8x$

$latex -6x\geq 24$

**Step 3:** Now, we divide by -6:

$latex \frac{-6}{-6}x\geq\frac{24}{-6}$

$latex x\leq -4$

## Inequalities – Practice problems

Put into practice what you have learned about inequalities to solve the following problems. Choose an answer and check it to see that you selected the correct one.

## See also

Interested in learning more about inequalities? Take a look at these pages:

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