Solving inequalities

The process to solve inequalities is the same as the process to solve equations, which uses inverse operations to keep the equation or inequality balanced. Instead of using an equals sign, however, the inequality symbol is used throughout.

Example

Solve the inequality 3m + 2 \textgreater 8.

The inequality will be solved when m is isolated on one side of the inequality. This can be done by using inverse operations at each stage of the process.

\begin{array}{rcl} 3m + 2 & \textgreater & -4 \\ -2 && -2 \\ 3m & \textgreater & -6 \\ \div 3 && \div 3 \\ m & \textgreater & -2 \end{array}

The final answer is m \textgreater -2, which means m can be any value that is bigger than -2, not including -2 itself. If this answer was to be placed on a number line, an open circle would be needed at -2 with a line indicating the numbers that are greater than 2.

Number line showing that m is greater than -2
Question

Solve the inequality 2(2c + 2) \leq 5. Show the answer on a number line.

2(2c + 2) \leq 5

Expand the bracket:

4c + 4 \leq 5

Subtract 4 from each side:

4c \leq 1

Divide each side by 4:

c \leq \frac{1}{4}

c \leq \frac{1}{4} is the final answer.

To show this answer on a number line, put a closed circle at \frac{1}{4} and indicate the numbers that are less than \frac{1}{4}.

Number line showing that c is less than 1/4

Extra care should be taken when the unknown in an inequality has a negative coefficient. Use an inverse operation to make the coefficient of the unknown positive.

Question

Solve the inequality 3(3 - x) \textless 6. Show the answer on a number line.

3(3 - x) \textless 6

Expand the bracket:

9 - 3x \textless 6

The unknown has a coefficient of -3, so now add 3x to both sides of the inequality:

9 \textless 3x +6

Subtract 6 from each side:

3 \textless 3x

Now divide both sides by 3:

1 \textless x, which can also be written as x \textgreater 1 (reading from right to left).

To show this answer on a number line, put an open circle at 1 and indicate the numbers greater than 1.

Number line showing that x is greater than 1