Question

Which statements are true when verifying the solution set of |6-x/3|>18 as x <-36 or x > 72?

Answer 1

Since, you have not mentioned the statements, but I am solving the expression as well as verifying which anyways may be able to make you understand the concept.

Answer:

Both $x<-36\quad \mathrm{or}\quad \:x>72$ are the True solutions.

Step-by-step explanation:

Considering the expression

$\left|6-\frac{x}{3}\right|>18$

$\mathrm{Apply\:absolute\:rule}:\quad \mathrm{If}\:|u|\:>\:a,\:a>0\:\mathrm{then}\:u\:<\:-a\:\quad \mathrm{or}\quad \:u\:>\:a$

$6-\frac{x}{3}<-18\quad \mathrm{or}\quad \:6-\frac{x}{3}>18$

solving

$6-\frac{x}{3}<-18$

$6-\frac{x}{3}-6<-18-6$

$-\frac{x}{3}<-24$

$3\left(-\frac{x}{3}\right)<3\left(-24\right)$

$\mathrm{Multiply\:both\:sides\:by\:-1\:\left(reverse\:the\:inequality\right)}$

$\left(-x\right)\left(-1\right)>\left(-72\right)\left(-1\right)$

$x>72$

also solving

$6-\frac{x}{3}>18$

$6-\frac{x}{3}-6>18-6$

$-\frac{x}{3}>12$

$-x>36$

$\mathrm{Multiply\:both\:sides\:by\:-1\:\left(reverse\:the\:inequality\right)}$

$\left(-x\right)\left(-1\right)<36\left(-1\right)$

$x<-36$

$\mathrm{Combine\:the\:intervals}$

$x<-36\quad \mathrm{or}\quad \:x>72$

Verifying the solution:

Putting the value x < -36 in $\left|6-\frac{x}{3}\right|>18$

let suppose x = -37 which is < -36

$\left|6-\frac{x}{3}\right|>18$

$\left|6-\frac{\left(-37\right)}{3}\right|>18$

$\mathrm{Apply\:rule}\:-\left(-a\right)=a$

$=\left|6+\frac{37}{3}\right|$

$=\left|\frac{55}{3}\right|$

$\mathrm{Apply\:absolute\:rule}:\quad \left|a\right|=a,\:a\ge 0$

$\frac{55}{3}>18$

$\mathrm{Therefore,\:the\:solution\:is}$

$\mathrm{True}$

also putting the value x > 72

let suppose x = 73 which is > 72

$|6-\frac{\left(73\right)}{3}|>\:18$

$=\left|-\frac{55}{3}\right|$

$\mathrm{Apply\:absolute\:rule}:\quad \left|-a\right|=a$

$=\frac{55}{3}$

$\frac{55}{3}>18$

$\mathrm{Therefore,\:the\:solution\:is}$

$\mathrm{True}$

So, both $x<-36\quad \mathrm{or}\quad \:x>72$ are the True solutions.

Answer 2

Answer: 2, 3, 4, & 6.

Step-by-step explanation: edge 2021