Question

b. use a table to solve each inequality. then write a rule that describes how to use division to solve the inequalities.
i. $-2x < 4$ ii. $3 \geq -3x$
iii. $-2x < 8$ iv. $6 \geq -3x$

Explanation:

Step 1: Solve \( -2x < 4 \)

To solve for \( x \), divide both sides by \( -2 \). Remember, when dividing by a negative number, the inequality sign flips.

$$ \frac{-2x}{-2} > \frac{4}{-2} $$

Simplify both sides:

$$ x > -2 $$

Step 2: Solve \( 3 \geq -3x \)

Divide both sides by \( -3 \), and flip the inequality sign.

$$ \frac{3}{-3} \leq \frac{-3x}{-3} $$

Simplify:

$$ -1 \leq x \quad \text{or} \quad x \geq -1 $$

Step 3: Solve \( -2x < 8 \)

Divide both sides by \( -2 \), flipping the inequality sign.

$$ \frac{-2x}{-2} > \frac{8}{-2} $$

Simplify:

$$ x > -4 $$

Step 4: Solve \( 6 \geq -3x \)

Divide both sides by \( -3 \), flipping the inequality sign.

$$ \frac{6}{-3} \leq \frac{-3x}{-3} $$

Simplify:

$$ -2 \leq x \quad \text{or} \quad x \geq -2 $$

Rule for Division in Inequalities:

When dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed (flipped) to maintain the truth of the inequality. If dividing by a positive number, the inequality sign remains the same.

Answer:

I. \( x > -2 \)
II. \( x \geq -1 \)
III. \( x > -4 \)
IV. \( x \geq -2 \)

Rule: When dividing both sides of an inequality by a negative number, reverse the inequality sign. If dividing by a positive number, keep the sign as is.