Question

which number line shows the solution to (\frac{2}{3}x - 5 > 3)? four number lines: 1. from -6 to 0, red arrow left of open circle at -3. 2. from 9 to 15, red arrow left of open circle at 12. 3. from 9 to 15, red arrow right of open circle at 12. 4. from -6 to 0, red arrow right of open circle at -3.

Explanation:

Step1: Solve the inequality for x

Start with the inequality \(\frac{2}{3}x - 5>3\). First, add 5 to both sides:
\(\frac{2}{3}x-5 + 5>3 + 5\)
Simplify to get \(\frac{2}{3}x>8\).

Step2: Isolate x

Multiply both sides by \(\frac{3}{2}\) (the reciprocal of \(\frac{2}{3}\)) to solve for x:
\(x>8\times\frac{3}{2}\)
Calculate \(8\times\frac{3}{2}\): \(8\div2 = 4\), \(4\times3=12\), so \(x > 12\).

Step3: Analyze the number lines

We need a number line with an open circle at 12 (since the inequality is \(>\), not \(\geq\)) and the arrow pointing to the right (because x is greater than 12). Looking at the options, the third number line (with 9, 10, 11, 12, 13, 14, 15 and the red arrow starting at an open circle at 12 and going right) matches this.

Answer:

The third number line (with numbers 9, 10, 11, 12, 13, 14, 15, open circle at 12, and red arrow pointing to the right)