2. Linear Equations and Inequalities

Absolute Value Inequalities

2. Linear Equations and Inequalities

Absolute Value Inequalities

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Multiple Choice

Solve and graph the following absolute value inequalities. Express the solution in interval notation.
(B) $\frac{1}{2} ∣ 2 x + 4 ∣ + 5 \geq 8$

$open bracket negative 5 comma 1 close bracket$

Number line from -8 to 8 with the solution to an absolute value inequality highlighted between -5 and 0 in red brackets.

$open bracket negative 1 comma 5 close bracket$

Number line from -8 to 8 with red brackets highlighting the interval between 0 and 5 on the x-axis.

$x is less than negative 1$ OR $x is greater than 5$

Number line graph showing solution intervals from negative infinity to -1 and from 5 to positive infinity, with open endpoints.

$x<-5$ OR $x>1$

Number line graph showing the solution to an absolute value inequality with intervals from negative infinity to -5 and from 1 to infinity.

Verified step by step guidance

Start with the given inequality: $\frac{1}{2} \left| 2x + 4 \right| + 5 \geq 8$.

Isolate the absolute value expression by subtracting 5 from both sides: $\frac{1}{2} \left| 2x + 4 \right| \geq 8 - 5$.

Simplify the right side: $\frac{1}{2} \left| 2x + 4 \right| \geq 3$.

Multiply both sides of the inequality by 2 to eliminate the fraction: $\left| 2x + 4 \right| \geq 6$.

Recall that for an absolute value inequality $|A| \geq B$ (where $B > 0$), the solution splits into two cases: $A \leq -B$ or $A \geq B$. Apply this to get two inequalities: \$2x + 4 \leq -6$ or $2x + 4 \geq 6$.

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Struggling with Intermediate Algebra?

Solve and graph the following absolute value inequalities. Express the solution in interval notation.(B) 12∣2x+4∣+5≥8\frac12\left|2x+4\right|+5\ge8

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Solve and graph the following absolute value inequalities. Express the solution in interval notation.
(B) $\frac{1}{2} ∣ 2 x + 4 ∣ + 5 \geq 8$