The Solution:
Given:
We are required to solve each of the above inequalities.
Apply the Modulus Inequality Theorem below:
[tex]\begin{gathered} \left|x\right|\le -9 \\ Absolute\text{ value cannot be less than 0.} \\ So,\left|x\right|\le-9\text{ has no solution.} \end{gathered}[/tex]
Thus, the correct answer is [option A]
For the inequality below:
[tex]\begin{gathered} \left|x\right|\le 9 \\ By\text{ the Absolute value rule:} \\ |u|\:\le \:a,\:a\:>\:0\:\mathrm{then}\:-a\:\le \:u\:\le \:a \end{gathered}[/tex][tex]\begin{gathered} \left|x\right|\le 9 \\ This\text{ becomes:} \\ -9\le \:x\le \:9 \\ Thus,\text{ the correct answer is \lbrack option B\rbrack} \end{gathered}[/tex]
For the third inequality:
[tex]\left|x\right|\ge -9[/tex]
Apply the rule below:
So,
[tex]\begin{gathered} \left|x\right|\ge -9 \\ \\ This\text{ becomes:} \\ x\leq-9 \\ or \\ x\ge9 \\ This\text{ means the solution is true for all values of x.} \end{gathered}[/tex]
Thus, the correct answer is [option C]
