How to Solve Linear Inequalities?

The process to solve linear inequalities in a variable is similar to solving basic equations. Most of the rules or techniques used to solve equations can be easily used to resolve inequalities. The only important difference is that the symbol of inequality changes direction when a negative number is multiplied or divided on both sides of an equation.

Here, we will look at examples with answers to dominate the process of solving linear inequalities.

ALGEBRA

Relevant for

Learning to solve linear inequalities with examples.

See examples

ALGEBRA

Relevant for

Learning to solve linear inequalities with examples.

See examples

Inequality symbols with illustrations

The following are the symbols used to represent inequalities:

Greater than

Symbol:

Example: x is greater than -2:

Graphic representation:

Less than

Symbol:

Example: x is less than 4:

Graphic representation:

Greater than or equal to

Symbol:

Example: x is greater than or equal to -3:

Graphic representation:

Less than or equal to

Symbol:

Example: x is less than or equal to 3:

Graphic representation:

Linear inequalities – Examples with answers

The following examples with answers will allow us to fully master the process of solving linear inequalities. Try to solve the exercises yourself before looking at the solution.

EXAMPLE 1

Solve and graph the solution of the inequality $latex 4x-6>2$.

To solve the inequality, we want to find all the x values that satisfy it. This means that there are an infinite number of solutions that, when substituted, would make the inequality true. Therefore, we follow the following steps:

• We write the original problem:

$latex 4x-6>2$

• We add 6 to both sides to keep the variables on one side and the constants on the other:

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

• After simplifying, the expression reduces to:

$latex 4x>8$

• We divide both sides by 4 and simplify to get the answer:

$latex \frac{4}{4}x>\frac{8}{4}$

$latex x>2$

• We use an open point to indicate that 2 is not part of the solution. The solution to the inequality includes all values to the right of 2, but does not include 2:

EXAMPLE 2

Solve and graph the inequality $latex -2x-3 \geq 1$.

This is an example of what happens to the inequality when we divide by a negative number:

• We write the original problem:

$latex -2x-3\geq 1$

• We isolate the variable by adding 3 to both sides:

$latex -2x-3+3\geq 1+3$

• After simplifying, the expression reduces to:

$latex -2x\geq 4$

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

$latex \frac{-2}{-2}x\geq \frac{4}{-2}$

$latex x\leq -2$

• We use a closed point to indicate that the -2 is part of the solution. The solution to the inequality includes -2 and all values to the left:

EXAMPLE 3

Solve the inequality $latex 5-3x>13-5x$.

Here, we have variables on both sides. It is possible to place the variables on the left side or on the right side, but the standard is to do it on the left side.

• We write the original problem:

$latex 5-3x>13-5x$

• We can add 5x to both sides and subtract 5 to solve for the variable:

$latex 5-3x+5x-5>13-5x+5x-5$

• Simplifying the expression, we have:

$latex 2x>8$

• We can get the answer by dividing both sides by 2:

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

$latex x> 2$

EXAMPLE 4

Solve inequality $latex 2(2x+3)<2x+12$.

In this case, we have parentheses, so we have to start by applying the distributive property to remove the parentheses:

• We write the original problem:

$latex 2(2x+3)<2x+12$

• We apply the distributive property:

$latex 2(2x)+2(3)<2x+12$

$latex 6x+6<2x+12$

• We isolate the variable by subtracting 6 and 2x from both sides:

$latex 6x+6-2x-6<2x+12-2x-6$

• After simplifying, the expression reduces to:

$latex 4x<8$

• We divide both sides by 4 and simplify to get the answer:

$latex \frac{4}{4}x< \frac{8}{4}$

$latex x< 2$

EXAMPLE 5

Solve the inequality $latex 2(3x-4)\geq 7x-7$.

Again, we start by applying the distributive property to remove the parentheses:

• We write the original problem:

$latex 2(3x-4)\geq 7x-7$

• We apply the distributive property:

$latex 2(3x)+2(-4)\geq 7x-7$

$latex 6x-8\geq 7x-7$

• Now, we add 8 and subtract 7x from both sides to isolate the variable:

$latex 6x-8+8-7x\geq 7x-7+8-7x$

• We simplify to obtain:

$latex -x\geq 1$

• We divide both sides by -1 and simplify to get the answer:

$latex \frac{-1}{-1}x\geq \frac{1}{-1}$

$latex x\geq -1$

EXAMPLE 6

Solve the inequality $latex 3(x+2)+2\leq 2(x-1)+8$.

In this case, we have to apply the distributive property to both parentheses and then combine like terms to simplify:

• We write the original problem:

$latex 3(x+2)+2\leq 2(x-1)+8$

• We apply the distributive property to both parentheses:

$latex 3(x)+3(2)+2\leq 2(x)+2(-1)+8$

$latex 3x+6+2\leq 2x-2+8$

• We combine like terms to simplify:

$latex 3x+8\leq 2x+6$

• We isolate the variable by subtracting 8 and 2x from both sides:

$latex 3x+8-8-2x\leq 2x+6-8-2x$

• Simplifying, we have:

$latex x\leq -2$

• We no longer have to divide:

$latex x\leq -2$

Linear inequalities – Practice problems

Test your skills and knowledge about linear inequalities with the following interactive problems. Choose an answer and check it to verify that you selected the correct answer. The examples solved above can serve as a guide if you have problems with these exercises.