# 2011 AMC 10A Problems/Problem 7

## Problem 7

Which of the following equations does NOT have a solution? $\text{(A)}\:(x+7)^2=0$ $\text{(B)}\:|-3x|+5=0$ $\text{(C)}\:\sqrt{-x}-2=0$ $\text{(D)}\:\sqrt{x}-8=0$ $\text{(E)}\:|-3x|-4=0$

## Solution 1 $|-3x|+5=0$ has no solution because absolute values output positives and this equation implies that the absolute value could output a negative.

Further: $(x+7)^2 = 0$ is true for $x = -7$ $\sqrt{-x}-2=0$ is true for $x = -4$ $\sqrt{x}-8=0$ is true for $x = 64$ $|-3x|-4=0$ is true for $x = \frac{4}{3}, -\frac{4}{3}$

Therefore, the answer is $\boxed{\mathrm{(B)}}$.

## Solution 2

Instead of solving, we can just categorize and solve.

Section 1: This contains A,C,D as they are all squares or square roots. From skimming, we can get an answer as maybe C

Section 2: This contains B and E From skimming we can get we can get answer as maybe B

Now we can analyze and we see $-x$ can become $x$ if $x=-y$ and absolute value inequalities cannot be negative, so the answer is $\boxed{\mathrm{(B)}}$

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 