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.

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1. Review of Real Numbers2h 43m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
44m

2. Linear Equations and Inequalities5h 35m

Solving Linear Equations
2h 23m

Linear Inequalities in One Variable
46m

Set Operations and Compound Inequalities
1h 7m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

3. Solving Word Problems2h 46m

Introduction to Problem Solving
37m

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25m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphs and Functions5h 12m

The Rectangular Coordinate System
44m

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24m

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23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

Linear Inequalities in Two Variables
28m

Introduction to Relations and Functions
53m

Function Notation
15m

Composition of Functions
17m

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Solving Systems of Linear Equations by Graphing
29m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

Systems of Linear Inequalities
23m

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The Product Rule
9m

The Quotient Rule
10m

Intro to the Power Rules
17m

The Power of a Quotient Rule
13m

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24m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

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20m

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38m

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34m

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38m

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11m

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37m

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41m

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41m

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42m

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25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

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27m

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Introduction to Radical Expressions
47m

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47m

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19m

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53m

Introduction to Complex Numbers
18m

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51m

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25m

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26m

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40m

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1h 28m

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25m

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33m

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32m

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35m

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41m

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Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
29m

Factorials
13m

2. Linear Equations and Inequalities

Absolute Value Inequalities

Intermediate Algebra

2. Linear Equations and Inequalities

Absolute Value Inequalities

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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.

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Problema precedenteProssimo problema

Guida verificata passo dopo passo

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.