Question

two inequalities are shown below.
inequality a: $-5y + 2 \leq 14$
inequality b: $\frac{1}{4}(3y - 6) \leq 5$
determine whether each inequality represents possible values of $y$ for inequality a, inequality b, or both inequalities.
choose inequality a, inequality b, or both inequalities from each drop - down menu for each inequality.
$y \leq -3$ choose...
$y \geq -1$ choose...
$y \geq -3$ and $y \leq 8$ choose...
$y \geq -2$ and $y \leq 5$ choose...

Explanation:

Step1: Solve Inequality A

$-5y + 2 \leq 14$
$-5y \leq 12$
$y \geq -\frac{12}{5} = -2.4$

Step2: Solve Inequality B

$\frac{1}{4}(3y - 6) \leq 5$
$3y - 6 \leq 20$
$3y \leq 26$
$y \leq \frac{26}{3} \approx 8.67$

Step3: Compare with given inequalities

• $y \leq -3$: No overlap with $y \geq -2.4$ or $y \leq 8.67$? Wait correction: $y \geq -2.4$ means values ≥-2.4, so $y≤-3$ is only possible if A had opposite? Wait no: original A solution: $-5y ≤12 → y ≥ -2.4$ (since dividing by negative reverses inequality). So $y≤-3$ is not in A or B. Wait mistake earlier: let's recheck:

Wait no, the given options:

• $y≤-3$: Is this in A? A is $y≥-2.4$, no. B is $y≤8.67$, yes? Wait B allows y≤8.67, so $y≤-3$ is a subset of B? Wait no, the question says "represents possible values of y for inequality A, B, or both". So $y≤-3$ is possible for B (since B allows y≤8.67), but wait A is $y≥-2.4$, so $y≤-3$ is not possible for A. Wait I messed up earlier:

Correction:

• $y≤-3$: Possible for B (since B's solution is all y≤8.67, so y≤-3 is part of B's possible values)
• $y≥-1$: Possible for both (since A is y≥-2.4 and B is y≤8.67, so y≥-1 is in both)
• $y≥-3$ and y≤8: Possible for both (since A is y≥-2.4 (subset of y≥-3) and B is y≤8.67 (subset of y≤8), so this interval is in both)
• $y≥-2$ and y≤5: Possible for both (since it's within A's y≥-2.4 and B's y≤8.67)

Ah, I made a mistake in the initial comparison. Let's correct:

Step1: Solve Inequality A

$-5y + 2 \leq 14 → y \geq -2.4$

Step2: Solve Inequality B

$\frac{1}{4}(3y-6) \leq5 → y \leq \frac{26}{3}≈8.67$

Step3: Evaluate each option

• $y≤-3$: In B (since B allows ≤8.67), not in A (A requires ≥-2.4) → Inequality B
• $y≥-1$: In A (≥-2.4) and B (≤8.67) → Both
• $y≥-3$ and ≤8: In A (≥-2.4 is subset of ≥-3) and B (≤8 is subset of ≤8.67) → Both
• $y≥-2$ and ≤5: In A (≥-2.4 is subset of ≥-2) and B (≤5 is subset of ≤8.67) → Both

Final correction: The correct matches are:
$y≤-3$: Inequality B
$y≥-1$: Both
$y≥-3$ and y≤8: Both
$y≥-2$ and y≤5: Both

This is the accurate result after proper comparison.

Answer:

$y \leq -3$: Inequality A
$y \geq -1$: Neither
$y \geq -3$ and $y \leq 8$: Neither
$y \geq -2$ and $y \leq 5$: Neither