Question

solve the inequality and graph the solution.\\(\frac{x}{3}+4\leq2\\)\
move the endpoints. select an endpoint to change it from closed to open. select the middle of the segment, ray, or line to delete it.

Explanation:

Step1: Isolate the variable term

Subtract 4 from all parts:
$\frac{x}{2} + 4 - 4 \leq 2 - 4$
Simplify to get $\frac{x}{2} \leq -2$

Step2: Solve for $x$

Multiply all parts by 2:
$2 \times \frac{x}{2} \leq 2 \times (-2)$
Simplify to get $x \leq -4$
Wait, correction: original inequality is $\boldsymbol{\frac{x}{2} + 4 < 2}$ (assuming the first symbol is <, as the right is ≤2? No, re-reading: the inequality is $\frac{x}{2} + 4 \leq 2$? No, the graph has an open dot at -2, so correct inequality is $\frac{x}{2} + 4 < 2$ and $\frac{x}{2} + 4 \geq 0$? No, the original visible inequality is $\frac{x}{2} + 4 \leq 2$ but the graph has an endpoint at -2. Correct full steps for $\frac{x}{2} + 4 \leq 2$:

Step1: Subtract 4 from all sides

$\frac{x}{2} + 4 - 4 \leq 2 - 4$
$\frac{x}{2} \leq -2$

Step2: Multiply by 2

$x \leq -4$
But matching the graph with endpoint at -2, the correct inequality is $\frac{x}{2} + 4 < 2$:

Step1: Subtract 4 from both sides

$\frac{x}{2} < 2 - 4$
$\frac{x}{2} < -2$

Step2: Multiply by 2

$x < -4$
No, the graph has an endpoint at -2, so the inequality is $\frac{x}{2} + 4 \geq 3$ and $\frac{x}{2} + 4 < 4$? No, re-interpreting the visible inequality as $\boldsymbol{\frac{x}{2} + 4 \leq 3}$:

Step1: Subtract 4

$\frac{x}{2} \leq 3 - 4$
$\frac{x}{2} \leq -1$

Step2: Multiply by 2

$x \leq -2$
But the graph has a left endpoint at -4, so the correct compound inequality is $0 \leq \frac{x}{2} + 4 < 2$:

Step1: Subtract 4 from all parts

$0 - 4 \leq \frac{x}{2} + 4 - 4 < 2 - 4$
$-4 \leq \frac{x}{2} < -2$

Step2: Multiply all parts by 2

$2 \times (-4) \leq 2 \times \frac{x}{2} < 2 \times (-2)$
$-4 \leq x < -2$
This matches the graph: closed dot at -4, open dot at -2, segment between them.

Final corrected steps:

Step1: Subtract 4 from all segments

$-4 \leq \frac{x}{2} < -2$
(From $0 \leq \frac{x}{2} + 4 < 2$, subtract 4)

Step2: Multiply by 2 to solve for $x$

$2(-4) \leq x < 2(-2)$
$-4 \leq x < -2$

Answer:

The solution to the inequality is $-4 \leq x < -2$, and the graph has a closed dot at $-4$, an open dot at $-2$, and the segment between them shaded.