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 1:
= Either
or Solve



Solve



The answers are 7.694933, -0.194933, 3.5 and 1. These answers may or may not be solutions to the original equation. You must verify each of the answers.



Check the solutions:
Check the answer x=7.694933 by substituting 7.694933 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 (not completely because we rounded 7.694933) to the right side of the original equation, the answer x=7.694933 is a solution to the original equation.


Check x=-0.194933 by substituting -0.194933 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 (not completely because we rounded -0.194933) to the right side of the original equation, the answer x=-0.194933 is a solution to the original equation.


Check x=3.5 by substituting 3.5 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, the answer x=3.5 is a solution to the original equation.


Check x=1 by substituting 1 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, the answer x=1 is a solution to the original equation.


The solutions are x=7.694933, -0.194933, 3.5, and 1.



You can also check your answer by graphing

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 the four x-intercepts on the graph are located at x=7.694933, -0.194933, 3.5, and 1. This verifies the four solutions by a graphical method.