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Solve the inequality $-\frac{1}{2}z\geq-1.74\text{.}$ 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 multiply by $-2$ to each side.

$\begin{aligned} -\frac{1}{2}z\amp\geq-1.74\\ \multiplyleft[]{(-2)}\left(-\frac{1}{2}z\right)\amp\highlight{{}\leq}\multiplyleft[]{(-2)}(-1.74)\\ z\amp\leq3.48 \end{aligned}$

In this exercise, we did multiply by a negative number and so the direction of the inequality sign changed.

Graphically, we represent this solution set as:

Using interval notation, we write the solution set as $(-\infty,3.48]\text{.}$ Using set-builder notation, we write the solution set as $\{z\mid z\leq3.48\}\text{.}$

We should check that $3.48$ is a solution, and also that some number less than $3.48$ is a solution.

$\begin{aligned} -\frac{1}{2}z\amp\geq-1.74\amp-\frac{1}{2}z\amp\geq-1.74\\ -\frac{1}{2}(\substitute{3.48})\amp\stackrel{?}{\geq}-1.74\amp-\frac{1}{2}(\substitute{0})\amp\stackrel{?}{\geq}-1.74\\ -1.74\amp\stackrel{\checkmark}{\geq}-1.74\amp0\amp\stackrel{\checkmark}{\geq}-1.74 \end{aligned}$

So our solution is reasonably checked.