Answer:
The solution to the inequality is:
[tex]-1\le \:x\le \:3[/tex]
The line graph of the solution is also attached.
Step-by-step explanation:
Given the expression
[tex]4x-1\:\le \:5x\:\le \:3\left(x+2\right)[/tex]
solving the expression
[tex]4x-1\:\le \:5x\:\le \:3\left(x+2\right)[/tex]
- [tex]\mathrm{If}\:a\le \:u\le \:b\:\mathrm{then}\:a\le \:u\quad \mathrm{and}\quad \:u\le \:b[/tex]
[tex]4x-1\le \:5x\quad \mathrm{and}\quad \:5x\le \:3\left(x+2\right)[/tex]
solving
[tex]4x-1\le \:5x[/tex]
Add 1 to both sides
[tex]4x-1+1\le \:5x+1[/tex]
Simplify
[tex]4x\le \:5x+1[/tex]
Subtract 5x from both sides
[tex]4x-5x\le \:5x+1-5x[/tex]
simplify
[tex]-x\le \:1[/tex]
Multiply both sides by -1 (reverse inequality)
[tex]\left(-x\right)\left(-1\right)\ge \:1\cdot \left(-1\right)[/tex]
Simplify
[tex]x\ge \:-1[/tex]
Similarly solving
[tex]5x\le \:3\left(x+2\right)[/tex]
Subtract 3x from both sides
[tex]5x-3x\le \:3x+6-3x[/tex]
Simplify
[tex]2x\le \:6[/tex]
Divide both sides by 2
[tex]\frac{2x}{2}\le \frac{6}{2}[/tex]
Simplify
[tex]x\le \:3[/tex]
So combine the interval
[tex]x\ge \:-1\quad \mathrm{and}\quad \:x\le \:3[/tex]
Merge overlapping intervals
[tex]-1\le \:x\le \:3[/tex]
Therefore, the solution to the inequality is:
[tex]-1\le \:x\le \:3[/tex]
The line graph of the solution is also attached.
