Manipulating and solving linear inequalities

Solving inequalities

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

Example

Solve the inequality \({3 m}~{+}~2~{\textgreater}~{-4}\)

The inequality will be solved when \({m}\) is isolated on one side of the inequality. This can be done by using inverse operations on each stage of the sum.

A solution to the inequality 3m + 2 > -4 by subtracting 2 from both sides then dividing both sides by 3

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 above this.

A number line from -5 to 5 with an empty circle over the -2 and an arrow pointing to 5.

As mentioned before, when manipulating inequalities the rules are exactly the same as when manipulating equations - but with one exception.

Example

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

Solution

First we expand the bracket:

(-2 x 2c) + (-2 x +2) ≥ -5

Then simplify:

-4c - 4 ≥ -5

Then we add 4 to both sides:

-4c ≥ -1

Finally, we divide both sides by -4 remembering that this changes the direction of the inequality:

\({c}~{\leq}~\frac {1}{4}\)

This can be shown on a number line as:

A number line from -5 to 5 with a closed circle over the 0 and an arrow pointing to -5.

Example

Solve the inequality \(\frac {3y}{2}\) + 4 > -5

First we subtract 4 from both sides:

\(\frac {3y}{2}\) > -9

Then we multiply both sides of the equation by 2

3y > -18

Finally we divide both sides of the equation by 3

y > -6

Question

Solve the inequality 3(3 - x) \({\textless}\) 6

Question

Solve the inequality \(\frac {2x}{5}\) - 7 ≥ 1

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Inequalities - Intermediate & Higher tier – WJECManipulating and solving linear inequalities