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Absolute Value Inequalities

Absolute Value Inequalities

Author: Sophia Tutorial
Description:

Solve an absolute value inequality.

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Tutorial

what's covered
  1. "Less Than" Absolute Value Inequalities
  2. "Greater Than" Absolute Value Inequalities
  3. Solving Absolute Value Inequalities
  4. No Solutions or All Real Solutions

1. "Less Than" Absolute Value Inequalities

When an inequality has an absolute value we will have to remove the absolute value in order to graph the solution or give interval notation. The way we remove the absolute value depends on the direction of the inequality symbol.

Consider vertical line x vertical line less than 2.

Absolute value is defined as distance from zero. Another way to read this inequality would be the distance from zero is less than 2. So on a number line we will shade all points that are less than 2 units away from zero.

File:5599-valueinequ1.png

This graph looks just like the graphs of the three part compound inequalities! When the absolute value is less than a number we will remove the absolute value by changing the problem to a three part inequality, with the negative value on the left and the positive value on the right. So vertical line x vertical line less than 2 becomes negative 2 less than x less than 2, as the graph above illustrates.

big idea
If the absolute value inequality is "less than" or "less than or equal to", we can write this as:
inequality: open vertical bar x close vertical bar less than a
rewrite as: short dash a less than x less than a
The inequality negative a less than x less than a is a type of "AND" compound inequality.
formula

Absolute Value Inequalities - Less Than
open vertical bar a x plus b close vertical bar less than c rightwards double arrow short dash c less than a x plus b less than c

2. "Greater Than" Absolute Value Inequalities

Consider vertical line x vertical line greater than 2.

Absolute value is defined as distance from zero. Another way to read this inequality would be the distance from zero is greater than 2. So on the number line we shade all points that are more than 2 units away from zero.

File:5600-valueinequ2.png

This graph looks just like the graphs of the OR compound inequalities! When the absolute value is greater than a number we will remove the absolute value by changing the problem to an OR inequality, the first inequality looking just like the problem with no absolute value, the second flipping the inequality symbol and changing the value to a negative. So vertical line x vertical line greater than 2 becomes x greater than 2 or x less than negative 2, as the graph above illustrates.

big idea
If the absolute value inequality is "greater than" or "greater than or equal to", we can write this as:
inequality: open vertical bar x close vertical bar greater than a
rewrite as: x greater than a space O R space x less than short dash a
This inequality x greater than a space O R space x less than negative a is a type of "OR" compound inequality.
formula

Absolute Value Inequalities - Greater Than
open vertical bar a x plus b close vertical bar greater than c rightwards double arrow a x plus b less than short dash c space O R space a x plus b greater than c space

3. Solving Absolute Value Inequalities

We can solve absolute value inequalities much like we solved absolute value equations by following these steps.

step by step

  1. Make sure the absolute value is isolated on one side.
  2. Remove the absolute value by either making a three-part inequality if the absolute value is less than a number, or making an OR inequality if the absolute value is greater than a number.
  3. Solve the inequality.
hint
Remember, if we multiply or divide by a negative the inequality symbol will switch directions!

EXAMPLE

Solve, graph, and give interval notation for the solution

open vertical bar 4 x minus 5 close vertical bar greater or equal than 6
Absolute value is greater, use OR
4 x minus 5 greater or equal than 6 space O R space 4 x minus 5 less or equal than short dash 6
Solve
stack plus 5 space space plus 5 with bar below space space space space space space space space space space space stack plus 5 space space space space space plus 5 with bar below
Add 5 to both sides
4 x greater or equal than 11 space O R space 4 x less or equal than short dash 1
Divide both sides by 4
stack space 4 space with bar on top space space stack space 4 space with bar on top space space space space space space space space space space space stack space 4 space with bar on top space space space space space space stack space 4 space with bar on top

x greater or equal than 11 over 4 space O R thin space x less or equal than short dash 1 fourth

Graph

open parentheses short dash infinity comma short dash 1 fourth text ] end text union left square bracket 11 over 4 comma infinity close parentheses Interval notation

hint
For all absolute value inequalities, we can also express our answers in interval notation which is done the same way it is done for standard compound inequalities.

EXAMPLE

Solve, graph, and give interval notation for the solution

short dash 4 minus 3 open vertical bar x close vertical bar less or equal than short dash 16
Add 4 to both sides to isolate the absolute value
stack plus 4 space space space space space space space space space space space space space space space space space plus 4 with bar below

short dash 3 open vertical bar x close vertical bar less or equal than short dash 12
Divide both sides by short dash 3
stack short dash 3 with bar on top space space space space space space space space space space stack short dash 3 with bar on top
Dividing by a negative switches the symbol
open vertical bar x close vertical bar greater or equal than 4

x greater or equal than 4 space O R space x less or equal than short dash 4
Absolute value is greater, use OR; Graph

open parentheses short dash infinity comma short dash 4 right square bracket space union space left square bracket 4 comma infinity close parentheses Interval notation

hint
In the previous example, we cannot combine −4 and−3 because they are not like terms, the − 3 has an absolute value attached. So we must first clear the − 4 by adding 4, then divide by −3. The next example is similar.

EXAMPLE


Solve, graph, and give interval notation for the solution

9 minus 2 open vertical bar 4 x plus 1 close vertical bar greater than 3
Subtract 9 from both sides
stack short dash 9 space space space space space space space space space space space space space space space space space space space short dash 9 with bar below

short dash 2 open vertical bar 4 x plus 1 close vertical bar greater than short dash 6
Divide both sides by short dash 2
stack space space space space space short dash 2 space space space space space with bar on top space space space space stack short dash 2 space with bar on top
Dividing by negative switches the symbol
open vertical bar 4 x plus 1 close vertical bar less than 3

short dash 3 less than 4 x plus 1 less than 3
Absolute value is less, use three part; Solve
stack negative 1 space space space space space minus 1 space space space minus 1 with bar below
Subtract 1 from all three parts
short dash 4 less than 4 x less than 2
Divide all three parts by 4
stack space 4 space with bar on top space space space stack space 4 space with bar on top space space space stack space 4 space with bar on top

short dash 1 less than x less than 1 half
Graph

open parentheses short dash 1 comma space 1 half close parentheses Interval Notation

hint
In the previous example, we cannot distribute the − 2 into the absolute value. We can never distribute or combine things outside the absolute value with what is inside the absolute value. Our only way to solve is to first isolate the absolute value by clearing the values around it, then either make a compound inequality (either a three-part inequality or an OR inequality) to solve.


4. No Solutions or All Real Solutions

It is important to remember as we are solving these equations, the absolute value is always positive. There are cases where there is no solution to the inequality or all real numbers are the solution:

  • No Solutions: If we end up with an absolute value is less than a negative number, then we will have no solution because absolute value will always be positive, greater than a negative.
  • All Real Solutions: If the absolute value is greater than a negative, this will always happen. Here the answer will be all real numbers.
Solve, graph, and give interval notation for the solution

12 plus 4 open vertical bar 6 x minus 1 close vertical bar less than 4
Subtract 12 from both sides
stack negative 12 space space space space space space space space space space space space minus 12 with bar below

4 open vertical bar 6 x minus 1 close vertical bar less than short dash 8
Divide both sides by 4
stack space space space 4 space space space with bar on top space space space space space space space space space stack space 4 space with bar on top

open vertical bar 6 x minus 1 close vertical bar less than short dash 2
Absolute value can't be less than a negative

N o space S o l u t i o n space o r space empty set Interval Notation

Solve, graph, and give interval notation for the solution

5 minus 6 open vertical bar x plus 7 close vertical bar less-than or slanted equal to 17
Subtract 5 from both sides
stack negative 5 space space space space space space space space space space space space space space space space space space space minus 5 with bar below

short dash 6 open vertical bar x plus 7 close vertical bar less-than or slanted equal to 12
Divide both sides by short dash 6
stack space short dash 6 space space space with bar on top space space space space stack short dash 6 with bar on top
Dividing by a negative flips the symbol
open vertical bar x plus 7 close vertical bar greater-than or slanted equal to short dash 2
Absolute value always greater than negative

A l l space R e a l space N u m b e r s space o r space straight real numbers

summary
"Less than" absolute value inequalities can be rewritten as "AND" compound inequalities, where our expression and our absolute value sign is bound between the negative and positive values of this quantity. "Greater than" absolute value inequalities can be written as "OR" compound inequalities, where your expression inside your absolute value sign is going to be less than the negative of this quantity, or it's going to be greater than the positive value of this quantity. When solving absolute value inequalities, remember to first isolate the absolute value, then remove the absolute value by either making a three-part inequality if the absolute value is less than a number, or making an OR inequality if the absolute value is greater than a number. There are also some instances where there is no solutions or all real solutions.

Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html

Formulas to Know
Absolute Value Inequalities - Greater Than

open vertical bar a x plus b close vertical bar greater than c rightwards double arrow a x plus b less than negative c space O R space a x plus b greater than c

Absolute Value Inequalities - Less Than

open vertical bar a x plus b close vertical bar less than c rightwards double arrow negative c less than a x plus b less than c