Question

4 ≤ y + 2 ≤ -3(y-2)+24​

Answer 1

Answer:

$4\le \:y+2\le \:-3\left(y-2\right)+24\quad :\quad \begin{bmatrix}\mathrm{Solution:}\:&\:2\le \:y\le \:7\:\\ \:\mathrm{Interval\:Notation:}&\:\left[2,\:7\right]\end{bmatrix}$

The solution graph is also attached.

Step-by-step explanation:

Given

$4\:\le \:y\:+\:2\:\le -3\left(y-2\right)+24$

as

$\mathrm{If}\:a\le \:u\le \:b\:\mathrm{then}\:a\le \:u\quad \mathrm{and}\quad \:u\le \:b$

so

$4\le \:y+2\quad \mathrm{and}\quad \:y+2\le \:-3\left(y-2\right)+24$

solving the intervals

as

$4\le \:y+2$

$y+2\ge \:4$

subtract 2 from both sides

$y+2-2\ge \:4-2$

$y\ge \:2$

also

$y+2\le -3\left(y-2\right)+24$

$y+2\le \:-3y+30$

subtract 2 from both sides

$y+2-2\le \:-3y+30-2$

$y\le \:-3y+28$

Add 3y to both sides

$y+3y\le \:-3y+28+3y$

$4y\le \:28$

Divide both sides by 4

$\frac{4y}{4}\le \frac{28}{4}$

$y\le \:7$

so combining the intervals

$y\ge \:2\quad \mathrm{and}\quad \:y\le \:7$

Merge overlapping intervals

$2\le \:y\le \:7$

Therefore,

$4\le \:y+2\le \:-3\left(y-2\right)+24\quad :\quad \begin{bmatrix}\mathrm{Solution:}\:&\:2\le \:y\le \:7\:\\ \:\mathrm{Interval\:Notation:}&\:\left[2,\:7\right]\end{bmatrix}$

The solution graph is also attached.