Question

solve the inequality. graph the solution. \\(|-6x + 8| > 7\\) select the correct choice below and, if necessary, fill in the answer box to complete your choice. \\(\bigcirc\\) a. the solution is \\(x < \square\\) or \\(x > \square\\). (type integers or fractions.) \\(\bigcirc\\) b. the solution is \\(\square < x < \square\\). (type integers or fractions.) \\(\bigcirc\\) c. the solution is all real numbers. \\(\bigcirc\\) d. there is no solution. choose the correct graph below. \\(\bigcirc\\) a. graph \\(\bigcirc\\) b. graph \\(\bigcirc\\) c. graph \\(\bigcirc\\) d. graph \\(\bigcirc\\) e. \\(\bigcirc\\) f. the graph contains no points.

Explanation:

Step1: Apply absolute value rule

For $|A|>B$, this splits into $A>B$ or $A<-B$.
So, $-6x + 8 > 7$ or $-6x + 8 < -7$

Step2: Solve first inequality

Isolate $x$ by subtracting 8, then dividing by -6 (reverse inequality sign).
$-6x > 7 - 8$
$-6x > -1$
$x < \frac{-1}{-6} = \frac{1}{6}$

Step3: Solve second inequality

Isolate $x$ by subtracting 8, then dividing by -6 (reverse inequality sign).
$-6x < -7 - 8$
$-6x < -15$
$x > \frac{-15}{-6} = \frac{5}{2}$

Step4: Match to graph

The solution $x < \frac{1}{6}$ or $x > \frac{5}{2}$ corresponds to a number line with open circles at $\frac{1}{6}$ and $\frac{5}{2}$, shading to the left of $\frac{1}{6}$ and right of $\frac{5}{2}$, which matches option A.

Answer:

A. The solution is $x < \frac{1}{6}$ or $x > \frac{5}{2}$.
A. <graph option A: left arrow from open circle left of 0, right arrow from open circle right of 0>