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:

Step1: Solve \(2x < 4\)

Divide both sides by 2 (positive, so inequality sign remains).
\(\frac{2x}{2} < \frac{4}{2}\)
\(x < 2\)

Step2: Solve \(3 \geq 3x\)

Divide both sides by 3 (positive, inequality sign remains).
\(\frac{3}{3} \geq \frac{3x}{3}\)
\(1 \geq x\) or \(x \leq 1\)

Step3: Solve \(2x < 8\)

Divide both sides by 2 (positive, inequality sign remains).
\(\frac{2x}{2} < \frac{8}{2}\)
\(x < 4\)

Step4: Solve \(6 \geq 3x\)

Divide both sides by 3 (positive, inequality sign remains).
\(\frac{6}{3} \geq \frac{3x}{3}\)
\(2 \geq x\) or \(x \leq 2\)

Step5: Write the rule

When dividing both sides of an inequality by a positive number, the direction of the inequality sign remains unchanged. For \(ax < b\) (or \(ax \leq b\), \(ax > b\), \(ax \geq b\)) where \(a > 0\), divide both sides by \(a\) to get \(x < \frac{b}{a}\) (or \(x \leq \frac{b}{a}\), \(x > \frac{b}{a}\), \(x \geq \frac{b}{a}\)).

Answer:

s (Solutions for Inequalities):
I. \(x < 2\)
II. \(x \leq 1\)
III. \(x < 4\)
IV. \(x \leq 2\)

Rule: When dividing both sides of an inequality by a positive number, the inequality sign does not change. For an inequality \(ax \, (\leq, <, \geq, >) \, b\) with \(a > 0\), solve by \(x \, (\leq, <, \geq, >) \, \frac{b}{a}\).