Answer :
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]
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]
