Question

1. solve each inequality for x and graph the solution. a. 6x - 20 > 4x - 8 b. 1/3(6x + 3) ≤ -2x

Explanation:

Step1: Solve part a

Subtract 4x from both sides.
\[6x - 4x-20>4x - 4x-8\]
\[2x-20>-8\]

Step2: Add 20 to both sides

\[2x-20 + 20>-8 + 20\]
\[2x>12\]

Step3: Divide both sides by 2

\[\frac{2x}{2}>\frac{12}{2}\]
\[x > 6\]
To graph, draw an open - circle at 6 and shade to the right.

Step4: Solve part b

Distribute the \(\frac{1}{3}\) on the left - hand side.
\[\frac{1}{3}(6x + 3)=2x + 1\]
The inequality becomes \(2x+1\leq - 2x\)

Step5: Add 2x to both sides

\[2x+2x + 1\leq-2x+2x\]
\[4x+1\leq0\]

Step6: Subtract 1 from both sides

\[4x+1 - 1\leq0 - 1\]
\[4x\leq - 1\]

Step7: Divide both sides by 4

\[\frac{4x}{4}\leq\frac{-1}{4}\]
\[x\leq-\frac{1}{4}\]
To graph, draw a closed - circle at \(-\frac{1}{4}\) and shade to the left.

Answer:

a. \(x>6\); Graph: Open - circle at 6, shade to the right.
b. \(x\leq-\frac{1}{4}\); Graph: Closed - circle at \(-\frac{1}{4}\), shade to the left.