SOLVING EQUATIONS 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 2:

Either tex2html_wrap_inline335 or

Step 1: tex2html_wrap_inline335 Solve

eqnarray30

eqnarray37

Step 2: tex2html_wrap_inline335 Solve

eqnarray51

eqnarray57

There are two real answers at and 0.219223593596,

and two imaginary answers at tex2html_wrap_inline347 and tex2html_wrap_inline349

Check the solution by substituting 2.2807764064 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 we substitute the value 2.2807764064 for x, then

is a solution.

Check the solution by substituting 0.219223593596 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 we substitute the value 0.219223593596 for x, then

is a solution.

Check the solution tex2html_wrap_inline347 by substituting tex2html_wrap_inline377 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 we substitute
the value tex2html_wrap_inline377

for x, then tex2html_wrap_inline347

is not a solution.

Check the solution tex2html_wrap_inline391 by substituting tex2html_wrap_inline393 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:

eqnarray221

Since the left side of the original equation is not equal to the right side of the original

equation after we substitute the value tex2html_wrap_inline393 for x, then tex2html_wrap_inline391 is not a solution.

The solutions are 1.73205080757 and -3.

You can also check the real answers by graphing

The graph is formed by subtracting the right side of the original equation from the left side of the original equation. Note that the graph has two x-intercepts located at 1.73205080757 and 1-3. This means that the two real solutions are 1.73205080757 and -3.

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.

If you would like to go back to the equation table of contents, click on Contents.

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

S.O.S. MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Author: Nancy Marcus