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Solve the inequality $x-5\gt-4\text{.}$ State the solution set using interval notation and using set-builder notation.

If needed, type inf or infinity for $\infty\text{,}$ >= for $\geq\text{,}$ and <= for $\leq\text{.}$

In interval notation, the solution set is

In set-builder notation, the solution set is

Solution

To solve this inequality, we will add $5$ to each side.

$\begin{aligned} x-5\amp\gt-4\\ x-5\addright{5}\amp\gt-4\addright{5}\\ x\amp\gt1 \end{aligned}$

Note again that the direction of the inequality did not change, since we did not multiply or divide each side of the inequality by a negative number at any point.

Graphically, we represent this solution set as:

Using interval notation, we write the solution set as $(1,\infty)\text{.}$ Using set-builder notation, we write the solution set as $\{x\mid x\gt1\}\text{.}$

We should check that $1$ is not a solution, but that some number greater than $1$ is a solution.

$\begin{aligned} x-5\amp\gt-4\amp x-5\amp\gt-4\\ \substitute{1}-5\amp\stackrel{?}{\lt}-4\amp \substitute{10}-5\amp\stackrel{?}{\lt}-4\\ -4\amp\stackrel{\text{no}}{\lt}-4\amp 5\amp\stackrel{\checkmark}{\lt}-4 \end{aligned}$

So our solution is reasonably checked.