Sure, let's solve each equation step-by-step to determine the number of solutions for each.
1. [tex]\(|2x + 1| = 0\)[/tex]:
- The absolute value function [tex]\(|a| = 0\)[/tex] is true if and only if [tex]\(a = 0\)[/tex].
- Therefore, for [tex]\(|2x + 1| = 0\)[/tex] to be true, we need:
[tex]\[
2x + 1 = 0
\][/tex]
- Solving for [tex]\(x\)[/tex]:
[tex]\[
2x + 1 = 0 \implies 2x = -1 \implies x = -\frac{1}{2}
\][/tex]
- Thus, this equation has one solution.
2. [tex]\(|x + 21| = 2\)[/tex]:
- The absolute value function [tex]\(|a| = b\)[/tex] (for [tex]\(b > 0\)[/tex]) has two solutions: [tex]\(a = b\)[/tex] or [tex]\(a = -b\)[/tex].
- Therefore, for [tex]\(|x + 21| = 2\)[/tex] to be true, we need:
[tex]\[
x + 21 = 2 \quad \text{or} \quad x + 21 = -2
\][/tex]
- Solving these two equations:
[tex]\[
x + 21 = 2 \implies x = 2 - 21 \implies x = -19
\][/tex]
[tex]\[
x + 21 = -2 \implies x = -2 - 21 \implies x = -23
\][/tex]
- Thus, this equation has two solutions.
3. [tex]\(2|x - 2| = -1\)[/tex]:
- The absolute value function [tex]\(|a|\)[/tex] is always non-negative ([tex]\(\geq 0\)[/tex]).
- Since the right side of the equation is [tex]\(-1\)[/tex] (a negative number), it is impossible for the absolute value function (which is always [tex]\(\geq 0\)[/tex]) to equal [tex]\(-1\)[/tex].
- Therefore, this equation has no solutions.
Matching the equations to the number of solutions:
- [tex]\(|2x + 1| = 0\)[/tex] ➜ one solution
- [tex]\(|x + 21| = 2\)[/tex] ➜ two solutions
- [tex]\(2|x - 2| = -1\)[/tex] ➜ no solutions
So the correct pairs would be as follows:
[tex]\[
\begin{array}{l}
|2 x+1|=0 \quad \text{➜} \quad \text{one solution} \\
|x + 21|=2 \quad \text{➜} \quad \text{two solutions} \\
2|x-2|=-1 \quad \text{➜} \quad \text{no solutions} \\
\end{array}
\][/tex]
