Question

1. $24 leq \frac{b}{4} + 5$
2. $-2.4r + 14.4 geq 2.4$
3. $-\frac{1}{6}(u + 1) > 1$

Explanation:

Problem 11: \( 24 \leq \frac{b}{4} + 5 \)

Step 1: Subtract 5 from both sides

To isolate the term with \( b \), we subtract 5 from both sides of the inequality.
\( 24 - 5 \leq \frac{b}{4} + 5 - 5 \)
\( 19 \leq \frac{b}{4} \)

Step 2: Multiply both sides by 4

To solve for \( b \), we multiply both sides of the inequality by 4.
\( 19 \times 4 \leq \frac{b}{4} \times 4 \)
\( 76 \leq b \) or \( b \geq 76 \)

Step 1: Subtract 14.4 from both sides

To isolate the term with \( r \), we subtract 14.4 from both sides of the inequality.
\( -2.4r + 14.4 - 14.4 \geq 2.4 - 14.4 \)
\( -2.4r \geq -12 \)

Step 2: Divide both sides by -2.4 (and reverse the inequality sign)

When dividing by a negative number, the inequality sign reverses.
\( \frac{-2.4r}{-2.4} \leq \frac{-12}{-2.4} \)
\( r \leq 5 \)

Step 1: Multiply both sides by -6 (and reverse the inequality sign)

When multiplying by a negative number, the inequality sign reverses.
\( -\frac{1}{6}(u + 1) \times (-6) < 1 \times (-6) \)
\( u + 1 < -6 \)

Step 2: Subtract 1 from both sides

To solve for \( u \), we subtract 1 from both sides of the inequality.
\( u + 1 - 1 < -6 - 1 \)
\( u < -7 \)

Answer:

\( b \geq 76 \)

Problem 12: \( -2.4r + 14.4 \geq 2.4 \)