Solve Inequality With Absolute Value

Learning Objectives

By the finish of this section, you lot volition be able to:

• Solve accented value equations
• Solve absolute value inequalities with "less than"
• Solve absolute value inequalities with "greater than"
• Solve applications with absolute value

Be Prepared 2.17

Earlier you go started, take this readiness quiz.

Evaluate: $\text{−}\left|\mathrm{seven}\right|.$
If yous missed this problem, review Example ane.12.

Be Prepared 2.18

Make full in $\text{<},\text{>},$ or $=$ for each of the following pairs of numbers.
ⓐ $\left|\mathrm{-8}\right|\text{\_\_\_}-\left|\mathrm{-eight}\right|$ ⓑ $12\text{\_\_\_}-\left|\mathrm{-12}\right|$ ⓒ $\left|\mathrm{-vi}\right|\text{\_\_\_}-\mathrm{six}$ ⓓ $\text{−}\left(\mathrm{-fifteen}\right)\text{\_\_\_}-\left|\mathrm{-fifteen}\right|$
If you missed this problem, review Example one.12.

Exist Prepared ii.xix

Simplify: $14-2\left|8-\mathrm{three}\left(4-1\right)\right|.$
If y'all missed this problem, review Example 1.13.

Solve Absolute Value Equations

As we prepare to solve accented value equations, nosotros review our definition of accented value.

Absolute Value

The accented value of a number is its distance from aught on the number line.

The absolute value of a number due north is written every bit $\left|\mathrm{north}\right|$ and $\left|n\right|\ge 0$ for all numbers.

Absolute values are always greater than or equal to goose egg.

We learned that both a number and its opposite are the aforementioned altitude from null on the number line. Since they have the same distance from nothing, they accept the aforementioned accented value. For example:

$\mathrm{-v}$ is v units away from 0, so $\left|\mathrm{-five}\right|=5.$

$\mathrm{five}$ is v units away from 0, so $\left|5\right|=5.$

Figure 2.6 illustrates this idea.

Effigy 2.6 The numbers 5 and $\mathrm{-v}$ are both five units abroad from goose egg.

For the equation $\left|x\right|=5,$ we are looking for all numbers that brand this a true argument. We are looking for the numbers whose distance from zero is 5. We just saw that both 5 and $\mathrm{-5}$ are v units from nothing on the number line. They are the solutions to the equation.

$$\begin{array}{cccccc}\text{If}\hfill & & & & & \left|\mathrm{10}\right|=5\hfill \\ \text{and then}\hfill & & & & & \mathrm{10}=\mathrm{-5}\text{or}\mathrm{ten}=\mathrm{five}\hfill \end{array}$$

The solution can be simplified to a single statement by writing $\mathrm{10}=\text{±}5.$ This is read, "x is equal to positive or negative 5".

We tin can generalize this to the following property for accented value equations.

Absolute Value Equations

For any algebraic expression, u, and any positive real number, a,

$$\begin{array}{cccccc}\text{if}\hfill & & & & & \left|u\right|=a\hfill \\ \text{then}\hfill & & & & & u=\text{−}a\text{or}u=a\hfill \end{array}$$

Remember that an accented value cannot be a negative number.

Example ii.68

Solve: ⓐ $\left|x\right|=\mathrm{eight}$ ⓑ $\left|y\right|=\mathrm{-6}$ ⓒ $\left|z\right|=0$

Attempt It 2.135

Solve: ⓐ $\left|\mathrm{10}\right|=2$ ⓑ $\left|y\right|=\mathrm{-4}$ ⓒ $\left|z\right|=0$

Try It 2.136

Solve: ⓐ $\left|x\right|=11$ ⓑ $\left|y\right|=\mathrm{-5}$ ⓒ $\left|z\right|=0$

To solve an absolute value equation, we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once nosotros isolate the absolute value expression we rewrite it as the two equivalent equations.

Instance 2.69

How to Solve Absolute Value Equations

Solve $\left|\mathrm{five}x-\mathrm{four}\right|-3=8.$

Try Information technology 2.137

Solve: $\left|3x-5\right|-\mathrm{one}=6.$

Try It 2.138

Solve: $\left|\mathrm{iv}x-3\right|-5=2.$

The steps for solving an accented value equation are summarized here.

How To

Solve absolute value equations.

1. Step 1. Isolate the absolute value expression.
2. Step 2. Write the equivalent equations.
3. Step 3. Solve each equation.
4. Step 4. Bank check each solution.

Example 2.70

Solve $2\left|\mathrm{ten}-7\right|+5=9.$

Try It 2.139

Solve: $3\left|x-\mathrm{iv}\right|-\mathrm{four}=8.$

Endeavor It 2.140

Solve: $2\left|x-\mathrm{v}\right|+3=9.$

Call back, an absolute value is e'er positive!

Example two.71

Solve: $\left|\frac{\mathrm{ii}}{\mathrm{three}}x-\mathrm{iv}\right|+\mathrm{xi}=3.$

Endeavour Information technology 2.141

Solve: $\left|\frac{\mathrm{three}}{\mathrm{iv}}x-5\right|+9=\mathrm{four}.$

Try It 2.142

Solve: $\left|\frac{5}{6}x+3\right|+8=\mathrm{vi}.$

Some of our absolute value equations could be of the course $\left|u\right|=\left|v\right|$ where u and five are algebraic expressions. For case, $\left|x-\mathrm{three}\right|=\left|2\mathrm{10}+\mathrm{one}\right|.$

How would we solve them? If 2 algebraic expressions are equal in accented value, then they are either equal to each other or negatives of each other. The belongings for absolute value equations says that for any algebraic expression, u, and a positive real number, a, if $\left|u\right|=a,$ and so $u=\text{−}a$ or $u=a.$

This tells united states of america that

$$\begin{array}{ccc}\text{if}\hfill & \left|u\right|=\left|v\right|\hfill & \\ \text{then}\hfill & u=\text{−}v\hfill & \text{or}u=v\end{array}$$

This leads us to the following property for equations with two accented values.

Equations with Two Absolute Values

For whatever algebraic expressions, u and five,

$$\begin{array}{cccccc}\text{if}\hfill & & & & & \left|u\right|=\left|v\right|\hfill \\ \text{then}\hfill & & & & & u=\text{−}v\text{or}u=v\hfill \end{array}$$

When nosotros accept the opposite of a quantity, nosotros must be careful with the signs and to add parentheses where needed.

Case 2.72

Solve: $\left|\mathrm{five}x-1\right|=\left|2x+3\right|.$

Endeavour It two.143

Solve: $\left|\mathrm{vii}x-\mathrm{three}\right|=\left|3\mathrm{ten}+7\right|.$

Try Information technology ii.144

Solve: $\left|\mathrm{half\; dozen}x-5\right|=\left|3\mathrm{ten}+4\right|.$

Solve Absolute Value Inequalities with "Less Than"

Let'southward look at present at what happens when we have an accented value inequality. Everything nosotros've learned most solving inequalities notwithstanding holds, but we must consider how the absolute value impacts our work.

Again nosotros will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line. For the equation $\left|x\right|=5,$ we saw that both 5 and $\mathrm{-5}$ are five units from zero on the number line. They are the solutions to the equation.

$$\begin{array}{c}\hfill \left|\mathrm{ten}\right|=5\hfill \\ \hfill \mathrm{ten}=\mathrm{-5}\text{or}x=5\hfill \end{array}$$

What about the inequality $\left|x\right|\le \mathrm{five}?$ Where are the numbers whose distance is less than or equal to 5? We know $\mathrm{-5}$ and 5 are both five units from nix. All the numbers between $\mathrm{-five}$ and five are less than v units from zero. Run across Figure two.7.

Effigy two.7

In a more general way, nosotros tin see that if $\left|u\right|\le a,$ so $\text{−}a\le u\le a.$ Come across Figure 2.8.

Figure 2.8

This result is summarized here.

Absolute Value Inequalities with $<$ or $\le$

For any algebraic expression, u, and whatever positive real number, a,

$$\begin{array}{cccccccccc}\text{if}\hfill & & & & \left|u\right|<a,\hfill & & & & & \text{then}\text{−}a<u<a\hfill \\ \text{if}\hfill & & & & \left|u\right|\le a,\hfill & & & & & \text{then}\text{−}a\le u\le a\hfill \end{array}$$

Later solving an inequality, it is often helpful to check some points to see if the solution makes sense. The graph of the solution divides the number line into 3 sections. Cull a value in each section and substitute it in the original inequality to run into if it makes the inequality truthful or not. While this is not a consummate check, it often helps verify the solution.

Case 2.73

Solve $\left|x\right|<\mathrm{seven}.$ Graph the solution and write the solution in interval notation.

Try It ii.145

Graph the solution and write the solution in interval notation: $\left|x\right|<9.$

Try It ii.146

Graph the solution and write the solution in interval note: $\left|x\right|<\mathrm{one}.$

Example two.74

Solve $\left|5x-\mathrm{vi}\right|\le \mathrm{four}.$ Graph the solution and write the solution in interval annotation.

Try It two.147

Solve $\left|\mathrm{ii}x-\mathrm{one}\right|\le \mathrm{v}.$ Graph the solution and write the solution in interval notation:

Effort Information technology 2.148

Solve $\left|\mathrm{iv}x-\mathrm{five}\right|\le 3.$ Graph the solution and write the solution in interval notation:

How To

Solve absolute value inequalities with < or ≤.

1. Step 1. Isolate the absolute value expression.
2. Step two. Write the equivalent compound inequality.
   

   $$\begin{array}{ccccccccccc}\left|u\right|<a\hfill & & & & & \text{is equivalent to}\hfill & & & & & \text{−}a<u<a\hfill \\ \left|u\right|\le a\hfill & & & & & \text{is equivalent to}\hfill & & & & & \text{−}a\le u\le a\hfill \end{array}$$

3. Pace 3. Solve the compound inequality.
4. Step 4. Graph the solution
5. Step v. Write the solution using interval notation.

Solve Absolute Value Inequalities with "Greater Than"

What happens for absolute value inequalities that have "greater than"? Over again nosotros will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line.

We started with the inequality $\left|\mathrm{ten}\right|\le 5.$ Nosotros saw that the numbers whose distance is less than or equal to five from zippo on the number line were $\mathrm{-5}$ and 5 and all the numbers between $\mathrm{-v}$ and 5. Run across Effigy ii.9.

Figure 2.9

Now we want to await at the inequality $\left|x\right|\ge 5.$ Where are the numbers whose distance from nix is greater than or equal to five?

Again both $\mathrm{-5}$ and v are five units from goose egg and so are included in the solution. Numbers whose distance from zero is greater than 5 units would be less than $\mathrm{-5}$ and greater than 5 on the number line. See Figure two.10.

Figure two.10

In a more general way, nosotros can see that if $\left|u\right|\ge a,$ then $u\le \text{−}a$ or $u\ge a.$ See Figure two.11.

Figure 2.11

This issue is summarized here.

Absolute Value Inequalities with > or ≥

For any algebraic expression, u, and any positive real number, a,

$$\begin{array}{ccccccccccc}\text{if}\hfill & & & & & \left|u\right|>a,\hfill & & & & & \text{then}u<-a\text{or}u>a\hfill \\ \text{if}\hfill & & & & & \left|u\right|\ge a,\hfill & & & & & \text{then}u\le \text{−}a\text{or}u\ge a\hfill \end{array}$$

Example ii.75

Solve $\left|\mathrm{ten}\right|>4.$ Graph the solution and write the solution in interval notation.

Try Information technology 2.149

Solve $\left|x\right|>2.$ Graph the solution and write the solution in interval notation.

Endeavor It ii.150

Solve $\left|\mathrm{10}\right|>\mathrm{i}.$ Graph the solution and write the solution in interval notation.

Example 2.76

Solve $\left|2x-3\right|\ge 5.$ Graph the solution and write the solution in interval notation.

Try Information technology two.151

Solve $\left|\mathrm{four}x-\mathrm{iii}\right|\ge 5.$ Graph the solution and write the solution in interval notation.

Attempt It 2.152

Solve $\left|3x-\mathrm{four}\right|\ge 2.$ Graph the solution and write the solution in interval notation.

How To

Solve accented value inequalities with > or ≥.

1. Footstep i. Isolate the absolute value expression.
2. Step 2. Write the equivalent compound inequality.
   

   $$\begin{array}{c}\left|u\right|>a\text{is equivalent to}u<-a\text{or}u>a\hfill \\ \left|u\right|\ge a\text{is equivalent to}u\le \text{−}a\text{or}u\ge a\hfill \end{array}$$

3. Step 3. Solve the compound inequality.
4. Stride 4. Graph the solution
5. Pace 5. Write the solution using interval note.

Solve Applications with Accented Value

Absolute value inequalities are often used in the manufacturing process. An detail must be made with near perfect specifications. Usually in that location is a sure tolerance of the difference from the specifications that is allowed. If the difference from the specifications exceeds the tolerance, the particular is rejected.

$$\left|\text{bodily-ideal}\right|\le \text{tolerance}$$

Example 2.77

The ideal diameter of a rod needed for a machine is 60 mm. The actual bore tin vary from the platonic diameter by $0.075$ mm. What range of diameters will be acceptable to the customer without causing the rod to exist rejected?

Try It ii.153

The ideal diameter of a rod needed for a machine is 80 mm. The actual diameter can vary from the ideal diameter past 0.009 mm. What range of diameters will be acceptable to the client without causing the rod to be rejected?

Attempt It 2.154

The ideal bore of a rod needed for a machine is 75 mm. The actual bore tin vary from the ideal diameter past 0.05 mm. What range of diameters volition be acceptable to the customer without causing the rod to be rejected?

Section 2.seven Exercises

Practice Makes Perfect

Solve Absolute Value Equations

In the following exercises, solve.

434 .

ⓐ $\left|x\right|=\mathrm{vi}$ ⓑ $\left|y\right|=\mathrm{-3}$ ⓒ $\left|z\right|=0$

435.

ⓐ $\left|\mathrm{10}\right|=4$ ⓑ $\left|y\right|=\mathrm{-five}$ ⓒ $\left|z\right|=0$

436 .

ⓐ $\left|x\right|=7$ ⓑ $\left|y\right|=\mathrm{-xi}$ ⓒ $\left|z\right|=0$

437.

ⓐ $\left|x\right|=3$ ⓑ $\left|y\right|=\mathrm{-1}$ ⓒ $\left|z\right|=0$

438 .

$\left|2x-3\right|-\mathrm{iv}=1$

439.

$\left|4\mathrm{10}-1\right|-3=0$

440 .

$\left|3x-\mathrm{iv}\right|+5=\mathrm{vii}$

441.

$\left|4x+\mathrm{vii}\right|+\mathrm{ii}=5$

442 .

$4\left|\mathrm{ten}-1\right|+\mathrm{ii}=\mathrm{x}$

443.

$3\left|x-\mathrm{iv}\right|+2=\mathrm{xi}$

444 .

$3\left|\mathrm{four}x-5\right|-\mathrm{iv}=11$

445.

$3\left|x+\mathrm{ii}\right|-5=4$

446 .

$\mathrm{-2}\left|\mathrm{10}-\mathrm{iii}\right|+8=\mathrm{-iv}$

447.

$\mathrm{-3}\left|\mathrm{ten}-4\right|+4=\mathrm{-5}$

448 .

$\left|\frac{\mathrm{three}}{4}x-3\right|+\mathrm{seven}=\mathrm{two}$

449.

$\left|\frac{\mathrm{three}}{\mathrm{five}}\mathrm{ten}-\mathrm{two}\right|+5=\mathrm{two}$

450 .

$\left|\frac{1}{2}\mathrm{ten}+5\right|+4=1$

451.

$\left|\frac{\mathrm{i}}{4}x+\mathrm{three}\right|+3=\mathrm{i}$

452 .

$\left|3x-2\right|=\left|2x-\mathrm{iii}\right|$

453.

$\left|\mathrm{four}\mathrm{ten}+3\right|=\left|2x+\mathrm{i}\right|$

454 .

$\left|6x-\mathrm{five}\right|=\left|\mathrm{ii}\mathrm{10}+3\right|$

455.

$\left|6-x\right|=\left|3-2x\right|$

Solve Absolute Value Inequalities with "less than"

In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.

460 .

$\left|3x-\mathrm{iii}\right|\le 6$

461.

$\left|2x-5\right|\le \mathrm{three}$

462 .

$\left|\mathrm{two}\mathrm{ten}+\mathrm{three}\right|+5<\mathrm{four}$

463.

$\left|3\mathrm{ten}-7\right|+3<1$

464 .

$\left|4\mathrm{ten}-3\right|<1$

465.

$\left|6x-5\right|<7$

466 .

$\left|\mathrm{10}-\mathrm{four}\right|\le \mathrm{-1}$

467.

$\left|5\mathrm{10}+1\right|\le \mathrm{-2}$

Solve Accented Value Inequalities with "greater than"

In the following exercises, solve each inequality. Graph the solution and write the solution in interval note.

472 .

$\left|3x-\mathrm{viii}\right|>\text{−}1$

473.

$\left|x-5\right|>\text{−}2$

474 .

$\left|\mathrm{iii}x-\mathrm{two}\right|>4$

475.

$\left|2\mathrm{ten}-\mathrm{ane}\right|>5$

476 .

$\left|x+\mathrm{three}\right|\ge 5$

477.

$\left|x-\mathrm{seven}\right|\ge 1$

478 .

$3\left|x\right|+\mathrm{iv}\ge \mathrm{one}$

479.

$5\left|x\right|+6\ge 1$

In the following exercises, solve. For each inequality, also graph the solution and write the solution in interval notation.

480 .

$\mathrm{ii}\left|\mathrm{10}+6\right|+4=8$

481.

$\left|6x-5\right|=\left|\mathrm{two}x+\mathrm{iii}\right|$

482 .

$\left|3\mathrm{ten}-4\right|\ge 2$

483.

$\left|\mathrm{two}x-\mathrm{five}\right|+2=\mathrm{three}$

484 .

$\left|4x-3\right|<5$

485.

$\left|3x+\mathrm{one}\right|-3=7$

486 .

$\left|\mathrm{seven}x+2\right|+\mathrm{viii}<4$

487.

$5\left|2\mathrm{10}-\mathrm{i}\right|-3=7$

488 .

$\left|8-x\right|=\left|4-\mathrm{iii}x\right|$

489.

$\left|x-\mathrm{vii}\right|>\text{−}3$

Solve Applications with Accented Value

In the post-obit exercises, solve.

490 .

A chicken farm ideally produces 200,000 eggs per day. But this total can vary by as much as 25,000 eggs. What is the maximum and minimum expected product at the farm?

491.

An organic juice bottler ideally produces 215,000 bottle per mean solar day. Only this full tin can vary by as much as 7,500 bottles. What is the maximum and minimum expected production at the bottling company?

492 .

In club to insure compliance with the law, Miguel routinely overshoots the weight of his tortillas past 0.5 gram. He just received a report that told him that he could be losing as much as $100,000 per yr using this practice. He at present plans to buy new equipment that guarantees the thickness of the tortilla inside 0.005 inches. If the platonic thickness of the tortilla is 0.04 inches, what thickness of tortillas will be guaranteed?

493.

At Lilly's Bakery, the ideal weight of a loaf of bread is 24 ounces. By law, the actual weight can vary from the ideal by one.v ounces. What range of weight will exist adequate to the inspector without causing the bakery being fined?

Writing Exercises

494 .

Write a graphical description of the absolute value of a number.

495.

In your own words, explain how to solve the accented value inequality, $\left|3x-2\right|\ge 4.$

Self Cheque

ⓐ Afterwards completing the exercises, use this checklist to evaluate your mastery of the objectives of this department.

ⓑ What does this checklist tell you about your mastery of this department? What steps will you take to ameliorate?

Solve Inequality With Absolute Value,

Source: https://openstax.org/books/intermediate-algebra-2e/pages/2-7-solve-absolute-value-inequalities

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