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 4: tex2html_wrap_inline209

Isolate the absolute value term by subtracting 2x from both sides of the equation.

Either tex2html_wrap_inline211 or tex2html_wrap_inline213

Solve tex2html_wrap_inline215

eqnarray39

eqnarray49

eqnarray61

Solve tex2html_wrap_inline213

eqnarray77

eqnarray87

eqnarray99

The answers are 1.06811457479, -0.468114574787, 2.04880884817 and -0.04880884817. 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 = 1.06811457479 by substituting 1.06811457479 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 1.06811457479 was substituted for x, the answer x=1.06811457479 is a solution to the original equation.

Check x=-0.468114574787 by substituting -0.468114574787 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 -0.468114574787 is substituted for x, the answer x=-0.468114574787 is a solution to the original equation.

Check x=2.04880884817 by substituting 2.04880884817 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 to the right side of the original equation after 2.04880884817 was substituted for x, the answer x=2.04880884817 is not a solution to the original equation.

Check x=-0.04880884817 by substituting -0.04880884817 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 -0.04880884817 was substituted for x, the answer x=-0.04880884817 is a solution to the original equation. Since we rounded the solution, the check will not be exact.

The solutions are x=1.06811457479,-0.468114574787, and -0.04880884817.

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 there are three x-intercepts on the graph are located at x=1.06811457479,-0.468114574787, and -0.04880884817. This verifies the three solutions by a graphical method.

If you would like to work another example, click on example.

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

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Author: Nancy Marcus