Question

solve and graph. $-6(4y - 3) \geq -102$
a. $y \geq -5$
number line from -10 to 10, closed dot at -5, shaded to the right
b. $y \geq 5$
number line from -10 to 10, closed dot at 5, shaded to the right
c. $y > 5$
number line from -10 to 10, open dot at 5, shaded to the right
d. $y \leq 5$
number line from -10 to 10, closed dot at 5, shaded to the left

Explanation:

Step1: Solve the inequality

Start with the inequality \(-6(4y - 3) \geq -102\). First, divide both sides by \(-6\). Remember that when dividing or multiplying an inequality by a negative number, the direction of the inequality sign flips. So we get:
\(4y - 3 \leq \frac{-102}{-6}\)
Simplify the right - hand side: \(\frac{-102}{-6}=17\)? Wait, no, \(-102\div(-6) = 17\)? Wait, no, \(-102\div(-6)=17\)? Wait, no, let's recalculate. \(-102\div(-6)\): 102 divided by 6 is 17, so \(-102\div(-6) = 17\)? Wait, no, the original inequality is \(-6(4y - 3)\geq - 102\). Divide both sides by - 6:
\(4y-3\leq\frac{-102}{-6}\)
\(\frac{-102}{-6} = 17\)? Wait, no, that can't be right. Wait, maybe I made a mistake. Wait, \(-6\times(4y - 3)\geq - 102\). Let's do it step by step.
First, expand the left - hand side: \(-24y+18\geq - 102\)
Then, subtract 18 from both sides: \(-24y\geq - 102 - 18\)
\(-24y\geq - 120\)
Now, divide both sides by - 24. Since we are dividing by a negative number, the inequality sign flips:
\(y\leq\frac{-120}{-24}\)
\(\frac{-120}{-24}=5\)
So the solution to the inequality is \(y\leq5\)

Step2: Analyze the graphs

• Option A: The inequality is \(y\geq - 5\), which does not match our solution \(y\leq5\).
• Option B: The inequality is \(y\geq5\), which does not match our solution.
• Option C: The inequality is \(y > 5\) (open circle, so strict inequality), which does not match our solution.
• Option D: The inequality is \(y\leq5\) (closed circle, and the shading is to the left of 5), which matches our solution.

Answer:

D. \(y\leq5\)