Question

1. graph the original inequality, solve it, and then graph its solution.

-4x ≤ -8

1. graph the original compound inequality, solve it, and graph each step of the solution process including the final answer.

-5 < 2/5x - 3 < 1

1. solve the compound inequality and match it with the graph of its solution. you will not use all the graphs provided.

a) -6 < -3x < 6
b) x + 5 ≤ 2 or 1/2x - 3 ≥ -2
c) -18 < 5x - 3 < 7

Explanation:

Step1: Solve the inequality \(-5 < \frac{2}{5}x - 3 < 1\)

Add \(3\) to all parts of the compound inequality:
\(-5+3 < \frac{2}{5}x - 3 + 3 < 1+3\)
\(-2 < \frac{2}{5}x < 4\)

Step2: Multiply all parts by \(\frac{5}{2}\)

\(-2\times\frac{5}{2}<\frac{2}{5}x\times\frac{5}{2}<4\times\frac{5}{2}\)
\(-5 < x < 10\)

Step3: Solve part (a) \(-6 < -3x < 6\)

Divide all parts by \(-3\) (and reverse the inequality signs):
\(\frac{-6}{-3}>\frac{-3x}{-3}>\frac{6}{-3}\)
\(2 > x > - 2\) or \(-2 < x < 2\)

Step4: Solve part (b) \(x + 5\leq2\) or \(\frac{1}{2}x-3\geq - 2\)

For \(x + 5\leq2\), subtract \(5\) from both sides: \(x\leq2 - 5\), \(x\leq - 3\)
For \(\frac{1}{2}x-3\geq - 2\), add \(3\) to both sides: \(\frac{1}{2}x\geq - 2 + 3\), \(\frac{1}{2}x\geq1\), multiply by \(2\): \(x\geq2\)

Step5: Solve part (c) \(-18 < 5x-3 < 7\)

Add \(3\) to all parts: \(-18 + 3<5x-3 + 3<7 + 3\)
\(-15 < 5x < 10\)
Divide by \(5\): \(\frac{-15}{5}<\frac{5x}{5}<\frac{10}{5}\)
\(-3 < x < 2\)

Answer:

1. For \(-4x\leq - 8\), divide by \(-4\) (reverse inequality sign) \(x\geq2\). The graph has a closed circle at \(2\) and a line to the right.
2. For \(-5 < \frac{2}{5}x - 3 < 1\), the solution is \(-5 < x < 10\). The graph has open circles at \(-5\) and \(10\) and a line between them.

3.

• For (a) \(-6 < -3x < 6\), the solution is \(-2 < x < 2\).
• For (b) \(x + 5\leq2\) or \(\frac{1}{2}x-3\geq - 2\), the solution is \(x\leq - 3\) or \(x\geq2\).
• For (c) \(-18 < 5x-3 < 7\), the solution is \(-3 < x < 2\).