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Solve for [tex]$x$: -2|2x + 3| \ \textgreater \ 4[/tex].

A. [tex]$x \ \textless \ -\frac{5}{2}$[/tex] or [tex][tex]$x \ \textgreater \ -\frac{1}{2}$[/tex][/tex]
B. Infinite number of solutions
C. No solution
D. [tex]$x \ \textless \ \frac{5}{2}$[/tex]

Sagot :

To solve the inequality [tex]\(-2|2x + 3| > 4\)[/tex], let's start by analyzing it step-by-step:

1. We have the inequality:
[tex]\[ -2|2x + 3| > 4 \][/tex]

2. Divide both sides by [tex]\(-2\)[/tex] and remember to flip the inequality sign since we are dividing by a negative number:
[tex]\[ |2x + 3| < -2 \][/tex]

3. Here, we are comparing the absolute value expression [tex]\(|2x + 3|\)[/tex] with [tex]\(-2\)[/tex]. However, the absolute value of any expression is always non-negative, meaning it is either zero or positive. Therefore, [tex]\(|2x + 3|\)[/tex] can never be less than a negative number:
[tex]\[ |2x + 3| \geq 0 \quad \text{for all real numbers } x \][/tex]

4. Since no real number [tex]\(x\)[/tex] can satisfy the inequality [tex]\(|2x + 3| < -2\)[/tex], it means there are no solutions to the given inequality.

So, the correct answer is:
[tex]\[ \text{No solution} \][/tex]