Module 3: The Absolute Value - Portland Community College



Section II: Functions, Inequalities, and the Absolute Value

[pic]

Module 3: The Absolute Value

The absolute value function is defined as follows:

[pic]

The graph of [pic] is shown in Figure 1 below.

[pic]

Figure 1: [pic]

[pic]

[pic] example: Solve [pic].

SOLUTION: Since both [pic] and [pic], both [pic] and [pic] are solutions, so the solution set is [pic].

[pic]

| |

|The Absolute-Value Principle for Equations |

| |

|For any positive number a and any algebraic expression X: |

| |

|▪ if [pic] then the equation [pic] has two solutions. |

| |

|▪ if [pic] then the equation [pic] has one solution. |

| |

|▪ if [pic] then the equation [pic] has no solutions. |

|[pic] |CLICK HERE FOR AN ABSOLUTE VALUE EQUATION EXAMPLE |

[pic]

|[pic] |CLICK HERE FOR ANOTHER ABSOLUTE VALUE EQUATION EXAMPLE |

[pic]

[pic] example: Solve [pic].

SOLUTION: Removing the absolute value symbols yields

[pic] or [pic]

which implies that

[pic] or [pic].

Thus, the solution set is [pic].

[pic]

[pic] example: Solve [pic].

SOLUTION: In order that [pic] we need [pic]. Thus, [pic] and the solution set is {–10}.

[pic]

[pic] example: Solve [pic] graphically.

SOLUTION: Below, we've graphed [pic] and [pic] (i.e., the left and right sides of the inequality). Since the absolute value function is below the horizontal line [pic] when [pic], the solution set for [pic] is [pic] which can be written in interval notation as [pic]. Another way to say this is that the y-values (i.e., the output values) of [pic] are less than [pic] when [pic], so the solution set for [pic] is the interval [pic].

[pic]

Figure 2: [pic] and [pic]

| |

|Principles for Solving Absolute Value Problems |

| |

|For any positive number a and any algebraic expression X : |

| |

|▪ the solutions of [pic] are those numbers that satisfy [pic]. |

| |

|▪ the solutions of [pic] are those numbers that satisfy [pic]. |

| |

|▪ the solutions of [pic] are those numbers that satisfy [pic]. |

|[pic] |CLICK HERE FOR AN ABSOLUTE VALUE INEQUALITY EXAMPLE |

[pic]

|[pic] |CLICK HERE FOR ANOTHER ABSOLUTE VALUE INEQUALITY EXAMPLE |

[pic]

[pic] example: Solve [pic].

SOLUTION: Removing the absolute value symbols yields

[pic] or [pic],

which can be simplified to

[pic] or [pic].

Thus, the solution set for [pic] is [pic]. We’ve graphed this set on the number line below.

[pic]

[pic]

[pic] Try these yourself and check your answers.

a. Solve [pic].

b. Solve [pic].

c. Solve [pic].

SOLUTIONS:

a.

[pic]

Thus, the solution set is {4, –1}.

b. The equation [pic] has no solutions since the absolute value of a number is never negative.

c.

[pic]

Thus, the solution set is [pic].

[pic]

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download