Question

when solving these inequalities, which solutions would you need to flip to make the inequality correct? *hint: there are two correct answers. □ $4(b - 7) < 22$ □ $17 - 3w \geq 35$ □ $5(12 - 3n) < 165$ □ $3a + 7 \leq 16$

Explanation:

To determine which inequalities require flipping the inequality sign, we recall that the inequality sign is flipped when we multiply or divide both sides by a negative number. Let's analyze each inequality:

1. Analyze \( 4(b - 7) < 22 \)

• Step 1: Divide both sides by 4 (a positive number).

\( b - 7 < \frac{22}{4} \) or \( b - 7 < 5.5 \).

• Step 2: Add 7 to both sides.

\( b < 5.5 + 7 \) or \( b < 12.5 \).
No sign flipping is needed (we divided by a positive number).

2. Analyze \( 17 - 3w \geq 35 \)

• Step 1: Subtract 17 from both sides.

\( -3w \geq 35 - 17 \) or \( -3w \geq 18 \).

• Step 2: Divide both sides by \( -3 \) (a negative number).

When dividing by a negative number, we flip the inequality sign:
\( w \leq \frac{18}{-3} \) or \( w \leq -6 \).
Sign flipping is needed.

3. Analyze \( 5(12 - 3n) < 165 \)

• Step 1: Divide both sides by 5 (a positive number).

\( 12 - 3n < \frac{165}{5} \) or \( 12 - 3n < 33 \).

• Step 2: Subtract 12 from both sides.

\( -3n < 33 - 12 \) or \( -3n < 21 \).

• Step 3: Divide both sides by \( -3 \) (a negative number).

Flip the inequality sign:
\( n > \frac{21}{-3} \) or \( n > -7 \).
Sign flipping is needed.

4. Analyze \( 3a + 7 \leq 16 \)

• Step 1: Subtract 7 from both sides.

\( 3a \leq 16 - 7 \) or \( 3a \leq 9 \).

• Step 2: Divide both sides by 3 (a positive number).

\( a \leq \frac{9}{3} \) or \( a \leq 3 \).
No sign flipping is needed (we divided by a positive number).

From the analysis, the inequalities that require sign flipping are \( \boldsymbol{17 - 3w \geq 35} \) and \( \boldsymbol{5(12 - 3n) < 165} \).

Final Answer:

The two inequalities that require flipping the sign are:

• \( 17 - 3w \geq 35 \)
• \( 5(12 - 3n) < 165 \)

Answer:

To determine which inequalities require flipping the inequality sign, we recall that the inequality sign is flipped when we multiply or divide both sides by a negative number. Let's analyze each inequality:

1. Analyze \( 4(b - 7) < 22 \)

• Step 1: Divide both sides by 4 (a positive number).

\( b - 7 < \frac{22}{4} \) or \( b - 7 < 5.5 \).

• Step 2: Add 7 to both sides.

\( b < 5.5 + 7 \) or \( b < 12.5 \).
No sign flipping is needed (we divided by a positive number).

2. Analyze \( 17 - 3w \geq 35 \)

• Step 1: Subtract 17 from both sides.

\( -3w \geq 35 - 17 \) or \( -3w \geq 18 \).

• Step 2: Divide both sides by \( -3 \) (a negative number).

When dividing by a negative number, we flip the inequality sign:
\( w \leq \frac{18}{-3} \) or \( w \leq -6 \).
Sign flipping is needed.

3. Analyze \( 5(12 - 3n) < 165 \)

• Step 1: Divide both sides by 5 (a positive number).

\( 12 - 3n < \frac{165}{5} \) or \( 12 - 3n < 33 \).

• Step 2: Subtract 12 from both sides.

\( -3n < 33 - 12 \) or \( -3n < 21 \).

• Step 3: Divide both sides by \( -3 \) (a negative number).

Flip the inequality sign:
\( n > \frac{21}{-3} \) or \( n > -7 \).
Sign flipping is needed.

4. Analyze \( 3a + 7 \leq 16 \)

• Step 1: Subtract 7 from both sides.

\( 3a \leq 16 - 7 \) or \( 3a \leq 9 \).

• Step 2: Divide both sides by 3 (a positive number).

\( a \leq \frac{9}{3} \) or \( a \leq 3 \).
No sign flipping is needed (we divided by a positive number).

From the analysis, the inequalities that require sign flipping are \( \boldsymbol{17 - 3w \geq 35} \) and \( \boldsymbol{5(12 - 3n) < 165} \).

Final Answer:

The two inequalities that require flipping the sign are:

• \( 17 - 3w \geq 35 \)
• \( 5(12 - 3n) < 165 \)