Question

directions: solve each inequality and graph th

1. 6(x - 1) ≤ -30

1. -2(4a + 4) + 5a > -35

1. 3k + 23 < 7k - 41

1. -2(5v - 12) > 6 - v

Explanation:

Problem 1: \( 6(x - 1) \leq -30 \)

Step 1: Divide both sides by 6

To isolate the term with \( x \), divide both sides of the inequality by 6.

$$ \frac{6(x - 1)}{6} \leq \frac{-30}{6} $$

Simplifying, we get:

$$ x - 1 \leq -5 $$

Step 2: Add 1 to both sides

To solve for \( x \), add 1 to both sides of the inequality.

$$ x - 1 + 1 \leq -5 + 1 $$

Simplifying, we get:

$$ x \leq -4 $$

Step 1: Distribute the -2

First, distribute the -2 across the terms inside the parentheses.

$$ -2 \cdot 4a + (-2) \cdot 4 + 5a > -35 $$

Simplifying, we get:

$$ -8a - 8 + 5a > -35 $$

Step 2: Combine like terms

Combine the \( a \)-terms:

$$ (-8a + 5a) - 8 > -35 $$

Simplifying, we get:

$$ -3a - 8 > -35 $$

Step 3: Add 8 to both sides

To isolate the term with \( a \), add 8 to both sides of the inequality.

$$ -3a - 8 + 8 > -35 + 8 $$

Simplifying, we get:

$$ -3a > -27 $$

Step 4: Divide by -3 (and reverse inequality)

Divide both sides by -3. Remember to reverse the inequality sign when dividing by a negative number.

$$ \frac{-3a}{-3} < \frac{-27}{-3} $$

Simplifying, we get:

$$ a < 9 $$

Step 1: Subtract \( 3k \) from both sides

To get all \( k \)-terms on one side, subtract \( 3k \) from both sides.

$$ 3k + 23 - 3k < 7k - 41 - 3k $$

Simplifying, we get:

$$ 23 < 4k - 41 $$

Step 2: Add 41 to both sides

To isolate the term with \( k \), add 41 to both sides.

$$ 23 + 41 < 4k - 41 + 41 $$

Simplifying, we get:

$$ 64 < 4k $$

Step 3: Divide by 4

Divide both sides by 4 to solve for \( k \).

$$ \frac{64}{4} < \frac{4k}{4} $$

Simplifying, we get:

$$ 16 < k $$

or

$$ k > 16 $$

Answer:

\( x \leq -4 \)

Problem 3: \( -2(4a + 4) + 5a > -35 \)