SOLVING EQUATIONS CONTAINING ABSOLUTE VALUE(S)

Note:

• if and only if
• if and only if a + b = 3 or a + b = -3
• Step1: Isolate the absolute value expression.

• Step 2: Set the quantity inside the absolute value notation equal to + and - the quantity on the other side of the equation.

• Step 3: Solve for the unknown in both equations.

Solve for x in the following equation.

Example 4 :

Step1: Isolate the absolute value expression by subtracting 5 from both sides of the equation.

Step 2: Set the quantity within the absolute value notation to .

Step 3: Solve for x in both equations:

The answers are 7 and . These answers may not be solutions to the equation. You must verify the answers.

Step 4: Check the solutions.

Check the answer x=7 by substituting 7 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer:

• Left side: .

• Right side: 8.5

Since the left side of the original equation equals the right side of the original equation after the value 7 is substituted for x, we have verified the solution x=7.

Check the answer x=-9.8 by substituting -9.8 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left side: .

• Right side: 8.5

Since the left side of the original equation equals the right side of the original equation after the value -9.8 is substituted for x, we have verified the solution =-9.8.

You can also check you answer by graphing (the left side of the original equation minus the right side of the original equation). You will note that the two x-intercepts on the graph are located at 7 and -9.8.

If you would like to test yourself by working some problems similar to this example, click on Problems .

[Algebra] [Trigonometry]
[Geometry] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra]