2 Answers

Mtbomb · Answer 1 · 2024-10-29T21:20:56+0000

Option #1: Solve the inequality and graph the solution set.

Option #2: Test out the options to see which has solutions.

Let's do Option #2:

The first possible answer shows t < -5, so if we pick t = -10, we'd be in that interval. Let's test if t = -10 is a solution:

$\begin{aligned}-8(-10) + 2(1)/(4) &\stackrel{?}{\geq} 42(1)/(4)\\[0.75em]80 + 2(1)/(4) &\stackrel{?}{\geq} 42(1)/(4)\\[0.75em]82(1)/(4) &\stackrel{\checkmark}{\geq} 42(1)/(4)\end{aligned}$

That checked!

So this alone tells us that the bottom two answers are not possible answers, since they do not include -10 in their sets.

But now we need to decide between the top answer or the one right below it. The only difference between the two number lines is that the top one does NOT include x = -5, while the second one does. So we need to test out if x = -5 is a solution, just like we did for x = -10.

$\begin{aligned}-8(-5) + 2(1)/(4) &\stackrel{?}{\geq} 42(1)/(4)\\[0.75em]40 + 2(1)/(4) &\stackrel{?}{\geq} 42(1)/(4)\\[0.75em]42(1)/(4) &\stackrel{\checkmark}{\geq} 42(1)/(4)\end{aligned}$

Because of the "or equal to" in the inequality, this does check.

Our answer is the second number line, with a closed circle at x = -5 and the shading to the left.

If you wanted to do this by solving the inequality, that work would look like this:

$\begin{aligned}-8t + 2(1)/(4) &\geq 42(1)/(4)\\[0.75em]-8t + 2(1)/(4)-2(1)/(4) &\geq 42(1)/(4)-2(1)/(4)\\[0.75em]-8t &\geq 40\\[0.75em](-8t)/(-8)~ &{\boldsymbol{\leq}}~ (40)/(-8)\\[0.75em]t &\leq -5\end{aligned}$