Question

mult./div. inequalities date: _____ period:

1. solve and graph each inequality.

a. ( \frac{1}{2}a < 20 ) b. ( -2 geq -\frac{d}{34} )
c. ( -72 < \frac{f}{6} ) d. ( -\frac{3}{4}g geq 12 )
e. ( 6p leq 96 ) f. ( 32 > -2y )
g. ( 4b leq -3 ) g. ( -6v > -72 )

Explanation:

Part a: Solve \(\boldsymbol{\frac{1}{2}a < 20}\)

Step1: Multiply both sides by 2

To isolate \(a\), we multiply both sides of the inequality by 2. Since 2 is positive, the inequality sign remains the same.
\(2\times\frac{1}{2}a < 2\times20\)

Step2: Simplify both sides

Simplifying the left side gives \(a\), and the right side gives 40.
\(a < 40\)

Step1: Multiply both sides by 34

Multiply both sides by 34 (positive, so inequality sign stays).
\(34\times(-2) \geq 34\times\frac{-d}{34}\)

Step2: Simplify both sides

Left side: \( -68 \geq -d\)

Step3: Multiply by -1 (reverse inequality)

Multiply both sides by -1. When multiplying by a negative number, reverse the inequality sign.
\((-1)\times(-68) \leq (-1)\times(-d)\)
\(68 \leq d\) or \(d \geq 68\)

Step1: Multiply both sides by 6

Multiply both sides by 6 (positive, inequality sign stays).
\(6\times(-72) < 6\times\frac{f}{6}\)

Step2: Simplify both sides

Left side: \(-432 < f\) or \(f > -432\)

Answer:

\(a < 40\) (Graph: Open circle at 40, arrow to the left)

Part b: Solve \(\boldsymbol{-2 \geq \frac{-d}{34}}\)