4.2A Solving Abs Value Eq/Inequalities

A Review of the Definition of Absolute Value

If |x| then x OR -x.

If |x + 1| then x + 1 OR -(x + 1).

If |2x -3| then 2x - 3 OR -(2x - 3).

Solving Absolute Value Equations

Solving absolute value equations and inequalities is different than simplifying absolute value expressions.

~ Set up the equation using the Absolute Value Definitions (above examples).

~ Eliminate the negative sign in front of the second expression.

~ Solve for x.

~ Determine if the values you found for x are the solutions to the original equation.

Note: Below these examples is an activity for you to work through and submit. You will be submitting work for the 4 absolute value inequalities. This will be checked in as a homework assignment worth 4 points.

Example 1 If |x + 2| = 5, find the value(s) of x

Solution

To solve this equation, use the algebraic definition of absolute value.

(x + 2) = 5 OR - (x + 2) = 5 Setup using the definition.

(x + 2) = 5 OR (x + 2) = -5 Divide both sides by -1.

Do NOT distribute the negative sign.

x = 5 -2 OR x = -5 -2 Solve for x.

x = 3 OR x = -7 Check if these values are solutions.

Check If x = 3, the absolute value equation is ... |3 + 2| = 5. True.

If x = -7, the absolute value equation is... |-7 + 2| = |-5| = 5. True.

The solution set to this absolute value equation is {3, -7}

All absolute value equations must include the word OR.

This equation is known as a disjunction. It is a compound sentence using the word OR.

Mathematically, if we have two mathematical statements, one or the other must be true to make the equation true.

Solving Absolute Value Inequalities

Absolute Value Inequalities

There are 4 inequalities used in Absolute Value Inequalities

< less than

≤ less than or equal to

> greater than

≥ greater than or equal to

Absolute Value Inequalities

Types of Absolute Inequalities

There are two types of absolute value inequalities:

Disjunctions and Conjunctions.

~ A disjunction must include the word OR.

Inequalities defined as disjunctions are > and ≥

Equations are also defined as disjunctions =

~ A conjunction must include the word AND.

Inequalities defined as conjunctions are < and ≤

An Important Reminder:

All absolute value equations and absolute value inequalities will have

TWO equations as a setup.

Example 1 If |x + 2| ≥ 5, find the value(s) of x
This problem is a disjunction, it includes greatOR than or equal to.
Solution
(x + 2) ≥ 5 OR -(x + 2) ≥ 5 We now need to solve for x.
(x + 2) ≥ 5 OR (x + 2) ≤ -5 Divide by -1, the inequality sign changed.
x ≥ 3 OR x ≤ -7
The solution set: (-∞, -7] U [3, ∞)

Example 2 If |x + 2 | ≤ 5, find the value(s) of x.

The problem is a conjunction, it includes less than or equal to.

Solution

(x + 2) ≤ 5 AND -(x + 2) ≤ 5 We now need to solve for x.

(x + 2) ≤ 5 AND x + 2 ≥ -5 Divide by -1, so the inequality sign changed.

x ≤ 3 AND x ≥ -7

The solutions set: [-7, 3]

Activity: Absolute Value Equations and Inequalities Act

~ It will be helpful if you arrange the problems on one side and the possible answers on the other side.

~ Do the work on loose-leaf paper and then drag the answer to the problem until it is connected.

~ You will submit your work for the 4 absolute value inequalities.

~ Save as: AbsIneq_LastName

~ Point value: 4 pts

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