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Solve Absolute Value Inequalities

Solve absolute value equations

Solve Absolute Value Equations

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

We learned that both a number and its opposite are the same distance from zero on the number line. Since they have the same distance from zero, they have the same absolute value. For example:

\(\ -5\) is 5 units away from 0, so \(|-5|=5.\)

\(\ 5\) is 5 units away from 0, so \(|5|=5.\)

illustrates this idea.

For the equation \(|x|=5,\) we are looking for all numbers that make this a true statement. We are looking for the numbers whose distance from zero is 5. We just saw that both 5 and \(-5\) are five units from zero on the number line. They are the solutions to the equation.

\[\begin{array}{llllll}\text{If} & & & & & |x|=5 \\ \text{then} & & & & & \ x=-5\ \text{or}\ x=5\end{array}\]

The solution can be simplified to a single statement by writing \(x=\text{\pm }5.\) This is read, “x is equal to positive or negative 5”.

Example

Try it.

Solve: ⓐ \(|x|=8\) ⓑ \(|y|=-6\) ⓒ \(|z|=0\)

Solution

ⓐ

	\(\|x\|=8\)
Write the equivalent equations.	\(x=-8\ \text{or}\ x=8\)
	\(x=\text{\pm }8\)

ⓑ

	\(\|y\|=-6\)
	No solution
Since an absolute value is always positive, there are no solutions to this equation.

ⓒ

	\(\|z\|=0\)
Write the equivalent equations.	\(z=-0\ \text{or}\ z=0\)
Since −0 = 0,	\(z=0\)
Both equations tell us that z = 0 and so there is only one solution.

How to Solve Absolute Value Equations

Try it.

Solve \(|5x-4|-3=8.\)

Solution

Example

Try it.

Solve \(2|x-7|+5=9.\)

Solution

	\(2\|x-7\|+5=9\)
Isolate the absolute value expression.	\(2\|x-7\|=4\)
	\(\|x-7\|=2\)
Write the equivalent equations.	\(\ x-7=\text{-}2\) or \(x-7=2\)
Solve each equation.	\(\ x=5\\) or \(\ x=9\)
Check:

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Solve Absolute Value Inequalities with “Less Than”

Let’s look now at what happens when we have an absolute value inequality. Everything we’ve learned about solving inequalities still holds, but we must consider how the absolute value impacts our work.

Again we 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 \(|x|=5,\) we saw that both 5 and \(-5\) are five units from zero on the number line. They are the solutions to the equation.

\[\begin{array}{l}|x|=5 \\ x=-5\ \text{or}\ x=5\end{array}\]

What about the inequality \(|x|\le 5?\) Where are the numbers whose distance is less than or equal to 5? We know \(-5\) and 5 are both five units from zero. All the numbers between \(-5\) and 5 are less than five units from zero. See .

In a more general way, we can see that if \(|u|\le a,\) then \(\text{-}a\le u\le a.\) See .

This result is summarized here.

Absolute Value Inequalities with

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

\[\begin{array}{llllllllll}\text{if} & & & & |u|

After 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 three sections. Choose a value in each section and substitute it in the original inequality to see if it makes the inequality true or not. While this is not a complete check, it often helps verify the solution.

Example

Try it.

Solve \(|x|<7.\) Graph the solution and write the solution in interval notation.

Solution


Write the equivalent inequality.
Graph the solution.
Write the solution using interval notation.

Check:

To verify, check a value in each section of the number line showing the solution. Choose numbers such as \(-8,\) 1, and 9.

Example

Try it.

Solve \(|5x-6|\le 4.\) Graph the solution and write the solution in interval notation.

Solution

Step 1. Isolate the absolute value expression. It is isolated.	\(\|5x-6\|\le 4\)
Step 2. Write the equivalent compound inequality.	\(-4\le 5x-6\le 4\)
Step 3. Solve the compound inequality.	\(2\le 5x\le 10\) \(\frac{2}{5}\le x\le 2\)
Step 4. Graph the solution.
Step 5. Write the solution using interval notation.	\([\frac{2}{5},2]\)
Check: The check is left to you.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Solve Absolute Value Inequalities with “Greater Than”

What happens for absolute value inequalities that have “greater than”? Again we 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 \(|x|\le 5.\) We saw that the numbers whose distance is less than or equal to five from zero on the number line were \(-5\) and 5 and all the numbers between \(-5\) and 5. See .

Now we want to look at the inequality \(|x|\ge 5.\) Where are the numbers whose distance from zero is greater than or equal to five?

Again both \(-5\) and 5 are five units from zero and so are included in the solution. Numbers whose distance from zero is greater than five units would be less than \(-5\) and greater than 5 on the number line. See .

In a more general way, we can see that if \(|u|\ge a,\) then \(u\le \text{-}a\) or \(u\ge a.\) See .

This result is summarized here.

Example

Try it.

Solve \(|x|>4.\) Graph the solution and write the solution in interval notation.

Solution

	\(\|x\|>4\)
Write the equivalent inequality.	\(x<-4\ \text{or}\ x>4\)
Graph the solution.
Write the solution using interval notation.	\((\text{-}\infty ,-4)\cup (4,\infty )\)
Check:

To verify, check a value in each section of the number line showing the solution. Choose numbers such as \(-6,\) 0, and 7.

Example

Try it.

Solve \(|2x-3|\ge 5.\) Graph the solution and write the solution in interval notation.

Solution

	\(\|2x-3\|\ge 5\)
Step 1. Isolate the absolute value expression. It is isolated.
Step 2. Write the equivalent compound inequality.	\(2x-3\le -5\ \text{or}\ 2x-3\ge 5\)
Step 3. Solve the compound inequality.	\(2x\le \text{-}2\ \text{or}\ 2x\ge 8\) \(x\le \text{-}1\ \text{or}\ x\ge 4\)
Step 4. Graph the solution.
Step 5. Write the solution using interval notation.	\((\text{-}\infty ,-1]\cup [4,\infty )\)
Check: The check is left to you.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Solve Applications with Absolute Value

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

\[|\text{actual-ideal}|\le \text{tolerance}\]

Example

Try it.

The ideal diameter of a rod needed for a machine is 60 mm. The actual diameter can vary from the ideal diameter by \(0.075\) mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

Solution

	Let x = the actual measurement.
Use an absolute value inequality to express this situation.	\(\|\text{actual-ideal}\|\le \text{tolerance}\)
	\(\|x-60\|\le 0.075\)
Rewrite as a compound inequality.	\(\text{-}0.075\le x-60\le 0.075\)
Solve the inequality.	\(59.925\le x\le 60.075\)
Answer the question.	The diameter of the rod can be between 59.925 mm and 60.075 mm.

Key Concepts

Absolute Value
The absolute value of a number is its distance from 0 on the number line.
The absolute value of a number n is written as \(|n|\) and \(|n|\ge 0\) for all numbers.
Absolute values are always greater than or equal to zero.
Absolute Value Equations
For any algebraic expression, u, and any positive real number, a,
\(\begin{array}{lllll}\text{if} & & & & |u|=a \\ \text{then} & & & & \ u=\text{-}a\ \text{or}\ u=a\end{array}\)
Remember that an absolute value cannot be a negative number.
How to Solve Absolute Value Equations
1. Isolate the absolute value expression.
2. Write the equivalent equations.
3. Solve each equation.
4. Check each solution.
Equations with Two Absolute Values
For any algebraic expressions, u and v,
\(\begin{array}{lllll}\text{if} & & & & |u|=|v| \\ \text{then} & & & & \ u=\text{-}v\ \text{or}\ u=v\end{array}\)
Absolute Value Inequalities with \(<\) or \(\le\)
For any algebraic expression, u, and any positive real number, a,
\(\begin{array}{lllllllllll}\text{if} & & & & & |u|
How To Solve Absolute Value Inequalities with \(<\) or \(\le\)
1. Isolate the absolute value expression.
2. Write the equivalent compound inequality.
  \(\begin{array}{lllllllllll}|u|
3. Solve the compound inequality.
4. Graph the solution
5. Write the solution using interval notation
Absolute Value Inequalities with \(>\) or \(\ge\)
For any algebraic expression, u, and any positive real number, a,
\(\begin{array}{lllllllllll}\text{if} & & & & & |u|>a, & & & & & \text{then}\ u<\text{-}a\ \text{or}\ u>a \\ \text{if} & & & & & |u|\ge a, & & & & & \text{then}\ u\le \text{-}a\ \text{or}\ u\ge a\end{array}\)
How To Solve Absolute Value Inequalities with \(>\) or \(\ge\)
1. Isolate the absolute value expression.
2. Write the equivalent compound inequality.
  \(\begin{array}{lllllllll}|u|>a & & & & \text{is equivalent to} & & & & u<\text{-}a\ \text{or}\ u>a \\ |u|\ge a & & & & \text{is equivalent to} & & & & u\le \text{-}a\ \text{or}\ u\ge a\end{array}\)
3. Solve the compound inequality.
4. Graph the solution
5. Write the solution using interval notation

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Evaluate: \(\text{-}|7|.\)
If you missed this problem, review .

Vastuse avalikustamine

\(-7\)
Fill in \(\text{<},\text{>},\) or \(=\) for each of the following pairs of numbers.
ⓐ \(|-8|\text{___}-|-8|\) ⓑ \(12\text{___}-|-12|\) ⓒ \(|-6|\text{___}-6\) ⓓ \(\text{-}(-15)\text{___}-|-15|\)
If you missed this problem, review .

Vastuse avalikustamine

a. >; b. >; c. >; d. >
Simplify: \(14-2|8-3(4-1)|.\)
If you missed this problem, review .

Vastuse avalikustamine

\(12\)
Solve: ⓐ \(|x|=8\) ⓑ \(|y|=-6\) ⓒ \(|z|=0\)

Vastuse avalikustamine

ⓐ
\(|x|=8\)
Write the equivalent equations. \(x=-8\ \text{or}\ x=8\)
\(x=\text{\pm }8\)
ⓑ
\(|y|=-6\)
No solution
Since an absolute value is always positive, there are no solutions to this equation.
ⓒ
\(|z|=0\)
Write the equivalent equations. \(z=-0\ \text{or}\ z=0\)
Since −0 = 0, \(z=0\)
Both equations tell us that z = 0 and so there is only one solution.
Solve: ⓐ \(|x|=2\) ⓑ \(|y|=-4\) ⓒ \(|z|=0\)

Vastuse avalikustamine

ⓐ \(\text{\pm }2\) ⓑ no solution ⓒ 0
Solve: ⓐ \(|x|=11\) ⓑ \(|y|=-5\) ⓒ \(|z|=0\)

Vastuse avalikustamine

ⓐ \(\text{\pm }11\) ⓑ no solution ⓒ 0
Solve \(|5x-4|-3=8.\)
Solve: \(|3x-5|-1=6.\)

Vastuse avalikustamine

\(x=4,x=-\frac{2}{3}\)
Solve: \(|4x-3|-5=2.\)

Vastuse avalikustamine

\(x=-1,x=\frac{5}{2}\)

Solve \(2|x-7|+5=9.\)

Vastuse avalikustamine

	\(2\|x-7\|+5=9\)
Isolate the absolute value expression.	\(2\|x-7\|=4\)
	\(\|x-7\|=2\)
Write the equivalent equations.	\(\ x-7=\text{-}2\) or \(x-7=2\)
Solve each equation.	\(\ x=5\\) or \(\ x=9\)
Check:

Solve: \(3|x-4|-4=8.\)

Vastuse avalikustamine

\(x=8,x=0\)
Solve: \(2|x-5|+3=9.\)

Vastuse avalikustamine

\(x=8,x=2\)
Solve: \(|\frac{2}{3}x-4|+11=3.\)

Vastuse avalikustamine

\(|\frac{2}{3}x-4|+11=3\)
Isolate the absolute value term. \(|\frac{2}{3}x-4|=-8\)
An absolute value cannot be negative. No solution
Solve: \(|\frac{3}{4}x-5|+9=4.\)

Vastuse avalikustamine

No solution
Solve: \(|\frac{5}{6}x+3|+8=6.\)

Vastuse avalikustamine

No solution

Solve: \(|5x-1|=|2x+3|.\)

Vastuse avalikustamine

	\(\|5x-1\|=\|2x+3\|\)
Write the equivalent equations. Solve each equation.	\(\begin{array}{llllllllll} & & & \ 5x-1 & = & \text{-}(2x+3) & \ \text{or}\ & 5x-1 & = & 2x+3 \\ & & & \ 5x-1 & = & -2x-3 & \ \text{or}\ & 3x-1 & = & 3 \\ & & & \ 7x-1 & = & -3 & & 3x & = & 4 \\ & & & \ 7x & = & -2 & & x & = & \frac{4}{3} \\ & & & \ x & = & -\frac{2}{7} & \ \text{or}\ & x & = & \frac{4}{3}\end{array}\)
Check.
We leave the check to you.

Solve: \(|7x-3|=|3x+7|.\)

Vastuse avalikustamine

\(x=-\frac{2}{5},\) \(x=\frac{5}{2}\)
Solve: \(|6x-5|=|3x+4|.\)

Vastuse avalikustamine

\(x=3,\) \(x=\frac{1}{9}\)
Solve \(|x|<7.\) Graph the solution and write the solution in interval notation.

Vastuse avalikustamine

Write the equivalent inequality.
Graph the solution.
Write the solution using interval notation.
Check:
To verify, check a value in each section of the number line showing the solution. Choose numbers such as \(-8,\) 1, and 9.
Graph the solution and write the solution in interval notation: \(|x|<9.\)

Vastuse avalikustamine
Graph the solution and write the solution in interval notation: \(|x|<1.\)

Vastuse avalikustamine

Solve \(|5x-6|\le 4.\) Graph the solution and write the solution in interval notation.

Vastuse avalikustamine

Step 1. Isolate the absolute value expression. It is isolated.	\(\|5x-6\|\le 4\)
Step 2. Write the equivalent compound inequality.	\(-4\le 5x-6\le 4\)
Step 3. Solve the compound inequality.	\(2\le 5x\le 10\) \(\frac{2}{5}\le x\le 2\)
Step 4. Graph the solution.
Step 5. Write the solution using interval notation.	\([\frac{2}{5},2]\)
Check: The check is left to you.

Solve \(|2x-1|\le 5.\) Graph the solution and write the solution in interval notation:

Vastuse avalikustamine
Solve \(|4x-5|\le 3.\) Graph the solution and write the solution in interval notation:

Vastuse avalikustamine
Solve \(|x|>4.\) Graph the solution and write the solution in interval notation.

Vastuse avalikustamine

\(|x|>4\)
Write the equivalent inequality. \(x<-4\ \text{or}\ x>4\)
Graph the solution.
Write the solution using interval notation. \((\text{-}\infty ,-4)\cup (4,\infty )\)
Check:
To verify, check a value in each section of the number line showing the solution. Choose numbers such as \(-6,\) 0, and 7.
Solve \(|x|>2.\) Graph the solution and write the solution in interval notation.

Vastuse avalikustamine
Solve \(|x|>1.\) Graph the solution and write the solution in interval notation.

Vastuse avalikustamine

Solve \(|2x-3|\ge 5.\) Graph the solution and write the solution in interval notation.

Vastuse avalikustamine

	\(\|2x-3\|\ge 5\)
Step 1. Isolate the absolute value expression. It is isolated.
Step 2. Write the equivalent compound inequality.	\(2x-3\le -5\ \text{or}\ 2x-3\ge 5\)
Step 3. Solve the compound inequality.	\(2x\le \text{-}2\ \text{or}\ 2x\ge 8\) \(x\le \text{-}1\ \text{or}\ x\ge 4\)
Step 4. Graph the solution.
Step 5. Write the solution using interval notation.	\((\text{-}\infty ,-1]\cup [4,\infty )\)
Check: The check is left to you.

Solve \(|4x-3|\ge 5.\) Graph the solution and write the solution in interval notation.

Vastuse avalikustamine
Solve \(|3x-4|\ge 2.\) Graph the solution and write the solution in interval notation.

Vastuse avalikustamine

Vastuse avalikustamine

	Let x = the actual measurement.
Use an absolute value inequality to express this situation.	\(\|\text{actual-ideal}\|\le \text{tolerance}\)
	\(\|x-60\|\le 0.075\)
Rewrite as a compound inequality.	\(\text{-}0.075\le x-60\le 0.075\)
Solve the inequality.	\(59.925\le x\le 60.075\)
Answer the question.	The diameter of the rod can be between 59.925 mm and 60.075 mm.

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

Vastuse avalikustamine

The diameter of the rod can be between 79.991 and 80.009 mm.
The ideal diameter of a rod needed for a machine is 75 mm. The actual diameter can vary from the ideal diameter by 0.05 mm. What range of diameters will be acceptable to the customer without causing the rod to be rejected?

Vastuse avalikustamine

The diameter of the rod can be between 74.95 and 75.05 mm.
\(|2x-3|-4=1\)
\(|4x-1|-3=0\)

Vastuse avalikustamine

\(x=1,x=-\frac{1}{2}\)
\(|3x-4|+5=7\)

Symbols used here

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

A \cup B,\ A \cap B,\ A \setminus B

union, intersection, difference
In either; in both; in A but not B.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

\neq

not equal
The two sides are different.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Solve Absolute Value Inequalities

Solve absolute value equations
Solve absolute value inequalities with “less than”
Solve absolute value inequalities with “greater than”
Solve applications with absolute value
Isolate the absolute value expression.
Write the equivalent equations.
Solve each equation.
Check each solution.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Proovi ise.

Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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	\(\|\frac{2}{3}x-4\|+11=3\)
Isolate the absolute value term.	\(\|\frac{2}{3}x-4\|=-8\)
An absolute value cannot be negative.	No solution