#### SOLVING EQUATION CONTAINING ABSOLUTE VALUE(S)

Note:

• if and only if

• if and only if a + b = 3 or a + b = -3

• if any only if a + b = +(x + y) or a + b = -(x + y)

Solve for x in the following equation.

Example 3:

Either or

Solve

Solve

The answers are 9.472135955, 0.527864045, 4.64575131106 and -0.64575131105. These answers may or may not be solutions to the original equation. You must verify each of the answers. In this case, two of the answers will be discarded as extraneous solutions.

Check the solutions:

Check the answer x=9.472135955 by substituting 9.472135955 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:

Since the left side of the original equation is equal to the right side of the original equation after 9.472135955 was substituted for x, the answer x=9.472135955 is a solution to the original equation.

Check x=0.527864045 by substituting 0.527864045 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:

Since the left side of the original equation is not equal to the right side of the original equation after 0.527864045 is substituted for x, the answer x=0.527864045 is not a solution to the original equation.

Check x=4.64575131106 by substituting 4.64575131106 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:

Since the left side of the original equation is equal to the right side of the original equation after 4.64575131106 was substituted for x, the answer x=4.64575131106 is a solution to the original equation.

Check x=-0.64575131105 by substituting -0.64575131105 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:

Since the left side of the original equation is not equal to the right side of the original equation after -0.64575131105 was substituted for x, , the answer x=-0.64575131105 is a not solution to the original equation. Since we rounded the solution, the check will not be exact.

The solutions are x=9.472135955 and 4.64575131106.

The graph was formed by subtracting the right side of the original equation from the left side of the original equation. You will note that there are two x-intercepts on the graph are located at x = 9.472135955 and 4.64575131106. This verifies the two solutions by a graphical method.

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