Exercises of linear inequalities can be solved by following a process similar to that used to solve linear equations. In general, the techniques used to solve linear equations are also useful for solving inequalities. The most important difference when solving inequalities is that when we divide or multiply the entire expression by a negative number, the inequality sign has to be switched.

The examples with answers that we will see will show the process of solving linear inequalities.

## Linear inequalities – Examples with answers

The following linear inequalities examples have their respective solution. The solution details the step-by-step process that can be followed to find the answer. It is recommended that you try to solve the exercises yourself before looking at the answer.

**EXAMPLE 1**

Solve and graph the inequality $latex 3x-5>1$.

##### Solution

- We start by writing the original problem:

$latex 3x-5>1$

- To solve for the variable, we add 5 to both sides of the inequality:

$latex 3x-5+5>1+5$

- After simplifying, the expression reduces to:

$latex 3x>6$

- To solve, we divide both sides by 3:

$latex \frac{3}{3}x> \frac{6}{3}$

$latex x> 2$

- We graph the inequality with an open point since 2 is not included in the solution. The solution is all the numbers to the right of 2:

**EXAMPLE 2**

Solve and graph the inequality $latex -4x+6<2$.

##### Solution

In this exercise, we will look at what happens to the inequality when we divide by a negative number:

- We have the inequality:

$latex -4x+6<2$

- We subtract 6 from both sides to solve for the variable:

$latex -4x+6-6<2-6$

- After simplifying, the expression reduces to:

$latex -4x<-4$

- Now, we have to divide by -4 to get the answer:

$latex \frac{-4}{-4}x< \frac{-4}{-4}$

$latex x>1$

Always remember that when we divide or multiply by a negative number, we change the inequality sign. |

- The 1 is not part of the solution, so we use an open point to indicate the solutions on the right side of the 1:

**EXAMPLE 3**

Solve and graph the inequality $latex 4x+2\geq 2x+10$.

##### Solution

In this case, we have variables on both sides. We have to move the variables to one side and the constants to the other. It doesn’t matter which side contains the variables, but it is common to move the variables to the left:

- We start with the original problem:

$latex 4x+2\geq 2x+10$

- We subtract 2 and 2x from both sides to solve for the variable:

$$4x+2-2-2x\geq 2x+10-2-2x$$

- Simplifying the inequality, we have:

$latex 2x\geq 8$

- We divide both sides by 2 and simplify to get the answer:

$latex \frac{2}{2}x\geq \frac{8}{2}$

$latex x\geq 4$

- Here, the 4 is part of the solution, so we use a closed point to indicate this:

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**EXAMPLE 4**

Solve the inequality $latex 2x+4<5x+10$.

##### Solution

We have to move the variables to one side of the inequality and the constants to the other:

- We have:

$latex 2x+4<5x+10$

- We isolate the variable by subtracting 4 and 5x from both sides :

$$2x+4-4-5x<5x+10-4-5x$$

- By simplifying, we get:

$latex -3x< 6$

- We have to divide both sides by -3 to get the answer:

$latex \frac{-3}{-3}x< \frac{6}{-3}$

$latex x>-2$

Always remember to change the direction of the inequality when dividing or multiplying by a negative number. |

**EXAMPLE 5**

Solve the inequality $latex 2(3x-3)>4x$.

##### Solution

In this case, we have parentheses, so we use the distributive property to remove parentheses and simplify:

- We write the original problem:

$latex 2(3x-3)>4x$

- We apply the distributive property:

$latex 2(3x)+2(-3)>4x$

$latex 6x-6>4x$

- We add 6 from both sides and subtract 4x to solve for the variable:

$$6x-6+6-4x>4x+6-4x$$

- After simplifying, the expression reduces to:

$latex 2x>6$

- By dividing both sides by 2, we have:

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

$latex x> 3$

**EXAMPLE 6**

Solve the inequality $latex 4(2x+5)<2(-x-4)-2$.

##### Solution

Here, we have to remove the parentheses from both sides and combine like terms to simplify:

- We have:

$latex 4(2x+5)<2(-x-4)-2$

- We apply the distributive property to both sides and combine like terms:

$$4(2x)+4(5)<2(-x)+2(-4)-2$$

$latex 8x+20<-2x-8-2$

$latex 8x+20<-2x-10$

- We isolate the variable by subtracting 20 and adding 2x to both sides:

$$8x+20-20+2x<-2x-10-20+2x$$

- We simplify to obtain:

$latex 10x< -30$

- We divide both sides by 10 and simplify to get the answer:

$latex \frac{10}{10}x< \frac{-30}{10}$

$latex x< -3$

**EXAMPLE 7**

Solve the inequality $latex 2(x+5)-10>4(2x-4)-2$.

##### Solution

Similar to the previous problem, we simplify the parentheses on both sides and combine like terms:

- We write the original problem:

$latex 2(x+5)-10>4(2x-4)-2$

- We apply the distributive property and combine like terms:

$$2(x)+2(5)-10>4(2x)+4(-4)-2$$

$latex 2x+10-10>8x-16-2$

$latex 2x>8x-18$

- We subtract 8x from both sides to solve for the variable:

$latex 2x-8x>8x-18-8x$

$latex 2x-8x>-18$

- Simplifying, we have:

$latex -6x>-18$

- We divide by -6 and simplify to get the answer:

$latex \frac{-6}{-6}x> \frac{-18}{-6}$

$latex x< 3$

Always remember to change the direction of the inequality when dividing or multiplying by a negative number. |

## Linear inequalities – Practice problems

Test your knowledge of linear inequalities with the following problems. Solve the inequalities and choose your answer. After clicking “Check”, you can check if you got the correct answer.

## See also

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

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