Question

for problems 18-26, solve and graph each inequality.

1. $-3(x + 6) < -21$
2. $60 < 6(x + 8)$
3. $-4(x + 5) < -28$

Explanation:

Problem 18: Solve \(-3(x + 6) < -21\)

Step 1: Divide both sides by -3 (reverse inequality)

When dividing an inequality by a negative number, the inequality sign flips. So we have:
\(x + 6 > \frac{-21}{-3}\)
Simplify the right side: \(\frac{-21}{-3}=7\), so \(x + 6 > 7\)

Step 2: Subtract 6 from both sides

Subtract 6 from each side to isolate \(x\):
\(x + 6 - 6 > 7 - 6\)
Simplify: \(x > 1\)

Step 1: Divide both sides by 6

\(\frac{60}{6} < x + 8\)
Simplify the left side: \(\frac{60}{6}=10\), so \(10 < x + 8\)

Step 2: Subtract 8 from both sides

\(10 - 8 < x + 8 - 8\)
Simplify: \(2 < x\) or \(x > 2\)

Step 1: Divide both sides by -4 (reverse inequality)

Dividing by a negative number flips the inequality sign:
\(x + 5 > \frac{-28}{-4}\)
Simplify the right side: \(\frac{-28}{-4}=7\), so \(x + 5 > 7\)

Step 2: Subtract 5 from both sides

\(x + 5 - 5 > 7 - 5\)
Simplify: \(x > 2\) Wait, no—wait, let's recheck:

Wait, \(-4(x + 5) < -28\)

Divide both sides by -4: \(x + 5 > \frac{-28}{-4}\) → \(x + 5 > 7\)

Subtract 5: \(x > 7 - 5\) → \(x > 2\)? Wait, no, wait: \(-4(x + 5) < -28\)

Wait, let's do it again:

\(-4(x + 5) < -28\)

Divide both sides by -4 (inequality flips): \(x + 5 > \frac{-28}{-4}\) → \(x + 5 > 7\)

Subtract 5: \(x > 7 - 5\) → \(x > 2\)? Wait, no, wait: \(-4(x + 5) = -4x -20\), so \(-4x -20 < -28\)

Add 20 to both sides: \(-4x < -8\)

Divide by -4 (flip inequality): \(x > 2\). Yes, that's correct.

Wait, but let's check with original: If \(x = 3\), \(-4(8) = -32 < -28\)? No, -32 is less than -28? Wait, -32 < -28 is true (since -32 is more negative). Wait, \(x = 2\): \(-4(7) = -28\), but the inequality is \(< -28\), so \(x = 2\) gives -28, which is not less than -28. So \(x > 2\) is correct.

Wait, no, wait: \(-4(x + 5) < -28\)

Let's plug \(x = 3\): \(-4(8) = -32 < -28\) → true.

\(x = 2\): \(-4(7) = -28\), which is not less than -28 → false. So \(x > 2\) is correct.

Answer:

\(x > 1\) (Graph: Open circle at 1, arrow to the right)

Problem 19: Solve \(60 < 6(x + 8)\)