Which of the following is the solution to [tex]$|x|-2 \leq-3$[/tex]?

A. No solution
B. All values are solutions
C. [tex][tex]$x \leq-1$[/tex][/tex]
D. [tex]$x \leq-1$ and x \geq 5$[/tex]



Answer :

To determine the solution to the inequality [tex]\( |x| - 2 \leq -3 \)[/tex], let's go through the steps:

1. We start with the inequality:
[tex]\[ |x| - 2 \leq -3 \][/tex]

2. Add 2 to both sides of the inequality to isolate the absolute value term:
[tex]\[ |x| - 2 + 2 \leq -3 + 2 \][/tex]
Simplifying this, we get:
[tex]\[ |x| \leq -1 \][/tex]

3. Now, we consider the property of absolute value. The absolute value [tex]\( |x| \)[/tex] of any real number [tex]\( x \)[/tex] is always non-negative, meaning [tex]\( |x| \)[/tex] is always greater than or equal to 0. Therefore, the inequality [tex]\( |x| \leq -1 \)[/tex] suggests that [tex]\( |x| \)[/tex] should be less than or equal to a negative number, which is impossible because absolute values are never negative.

Based on this reasoning, there is no real number [tex]\( x \)[/tex] that can satisfy the inequality [tex]\( |x| \leq -1 \)[/tex].

Thus, the solution to the inequality is:

A. No solution