Solve and graph the following absolute value inequalities. ( B ) ∣ 0.5 x + 1 ∣ + 8 ≥ 8 \left|0.5x+1\right|+8\ge8

Welcome back. In this problem, we'd like to solve the inequality, absolute value of 0.5 x plus one plus eight is greater than or equal to eight, and we'd like to graph our solution. Now our first step is going to be isolating the absolute value on the left side of the inequality, which means getting rid of this plus eight. We can do that by subtracting eight from both sides of the inequality, which will cancel with the plus eight on the left, leaving us with absolute value of 0.5 x plus one, is greater than or equal to eight minus eight, which is zero. Now, again, we know that the absolute value of something represents a distance, so it is always greater than or equal to zero no matter what we plug in for x. This means that this is always a true statement and that our solution set is all real numbers. We can represent all real numbers on the graph by tracing over the entire number line and then adding arrowheads pointing off in the direction of negative infinity to the left and positive infinity to the right. Again, because we end up simplifying our inequality down to a statement which is always true, we know that our solution set is all real numbers. Great work.

2. Linear Equations and Inequalities

Absolute Value Inequalities

Intermediate Algebra

2. Linear Equations and Inequalities

Absolute Value Inequalities

Struggling with Intermediate Algebra?

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Multiple Choice

Solve and graph the following absolute value inequalities.
(B) $∣ 0.5 x + 1 ∣ + 8 \geq 8$

All real numbers

Number line from -8 to 8 with a red arrow above pointing from left to right along the x-axis.

$x=2$

Number line graph showing the solution to the absolute value inequality |0.5x + 1| + 8 ≥ 8 with a red point at 2.

$x=-2$

Number line from -8 to 8 with a red point marked at -2 representing a solution to an absolute value inequality.

No solution

Verified step by step guidance

Start with the given inequality: $\left|0.5x + 1\right| + 8 \geq 8$.

Isolate the absolute value expression by subtracting 8 from both sides: $\left|0.5x + 1\right| \geq 8 - 8$ which simplifies to $\left|0.5x + 1\right| \geq 0$.

Recall that the absolute value of any real number is always greater than or equal to zero, so this inequality holds for all values of $x$.

Since the inequality is true for all $x$, the solution set is all real numbers.

To graph the solution, shade the entire number line because every value of $x$ satisfies the inequality.

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Struggling with Intermediate Algebra?

Solve and graph the following absolute value inequalities.(B) ∣0.5x+1∣+8≥8\left|0.5x+1\right|+8\ge8

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Solve and graph the following absolute value inequalities.
(B) $∣ 0.5 x + 1 ∣ + 8 \geq 8$