Solve the inequality $x + 7 < \frac{17 + 3x}{2} \le 15 - 3x$ and show your solution on a number line. Answer

1. Split the compound inequality into two separate inequalities.

From
$x+7<\frac{17+3x}{2}\le 15-3x,$
we solve
$x+7<\frac{17+3x}{2}$
and
$\frac{17+3x}{2}\le 15-3x.$

2. Solve the left-hand inequality.

$x+7<\frac{17+3x}{2}$
Multiply by $2$:
$2x+14<17+3x$
$14<17+x$
$-3<x$
So,
$x>-3.$

3. Solve the right-hand inequality.

$\frac{17+3x}{2}\le 15-3x$
Multiply by $2$:
$17+3x\le 30-6x$
$9x\le 13$
$x\le \frac{13}{9}.$

4. Combine both solution conditions and describe the number line representation.

The values of $x$ must satisfy both
$x>-3$
and
$x\le \frac{13}{9}.$
Hence,
$-3<x\le \frac{13}{9}.$
On the number line, use an open circle at $-3$, a closed circle at $\frac{13}{9}$, and shade the region between them.