Intermediate Algebra, Chapter 2, Test, Section Test, Problem 24

Evaluate the absolute value inequality $|7 - x| \leq -1$

By using the property of absolute value, we have

$
\begin{equation}
\begin{aligned}
7 - x &\leq -1 && \text{and} & 7 - x &\geq -(-1) \\
\\
-x &\leq -8 && \text{and} & -x &\geq - 6
&& \text{Subtract } 7\\
\\
x &\geq 8 && \text{and} & x &\leq 6
&& \text{Divide each side by $-1$. Remember that if you divide or multiply negative numbers ,the inequality symbol reverses}
\end{aligned}
\end{equation}
$

Since the inequalities are joined with $and$, then we are required to find the intersection of the two inequalities.
However both inequalities will not intersect. Thus, the solution is an empty set or $\cancel{0}$

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