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The water depth for a pool is set to 6ft, but the actual depth of the pool may vary as much as 4 inches. Write and solve absolute-value inequality to find the range of possible water depths in inches. Graph the solutions.

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1 Answer

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Here are some hints to get you started. Since you have measurements in both feet and inches, it’s convenient to convert everthing to inches: the depth is set to $6\cdot12=72$ inches but may vary up to $4$ inches either way.

Now recall that for any real numbers $a$ and $b$, $|a-b|$ is the distance between $a$ and $b$. If $x$ is the actual depth of the water, we’re told that its distance from $72$ inches is at most $4$ inches. Translate the words into symbols: the distance from $x$ to $72$ is $|x-72|$, and we’re told that this is at most $4$, so the inequality that you want is $|x-72|\le 4$.

The last major ingredient is knowing that the absolute value inequality $$|\text{thing}|\le u$$ translates to a pair of simultaneous inequalities that don’t involve absolute values: $$-u\le\text{thing}\le u\;.$$ Apply this to the inequality $|x-72|\le 4$ and solve the resulting pair of inequalities, and you’ll have all of the answer except the graph.

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