SOLVING EQUATIONS CONTAINING ABSOLUTE VALUE(S)

Note:


Solve for x in the following equation.

Problem 3.5b:

tex2html_wrap_inline196





Answer: tex2html_wrap_inline198 tex2html_wrap_inline198





Solution:

Either tex2html_wrap_inline198 tex2html_wrap_inline200 tex2html_wrap_inline198 ortex2html_wrap_inline198 tex2html_wrap_inline202





Step 1:tex2html_wrap_inline198 Solve tex2html_wrap_inline204

eqnarray32


eqnarray39





Step 2:tex2html_wrap_inline198 Solve tex2html_wrap_inline202

eqnarray52


eqnarray57





There are four real answers:. : tex2html_wrap_inline208 tex2html_wrap_inline210 However all these answers may or may not be solutions. You must verify that your answers are the solutions to the original equation.





Step 4:tex2html_wrap_inline198Check the solutions:


Check tex2html_wrap_inline212 by substituting tex2html_wrap_inline214 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.

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_inline214 for x, then tex2html_wrap_inline212 is not a solution.







Check the solution x=-2 by substituting -2 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.

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value -2 for x, then x=-2 is a solution.







Check the solution tex2html_wrap_inline236 by substituting 2.83205 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.

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value tex2html_wrap_inline244 for x, then tex2html_wrap_inline236 is a solution.







Check the solution tex2html_wrap_inline248 by substituting x=-0.73205 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.

Since the left side of the original equation is not equal to the right side of the original equation after we substitute the value 1- tex2html_wrap_inline258 for x, then tex2html_wrap_inline260 is not a solution.







There are two solutions: tex2html_wrap_inline262





You can also verify these two real answers by graphing

eqnarray156

The graph is formed by subtracting the right side of the original equation from the left side of the original equation. Note that there are x-intercepts at -2 and at 2.73205. This verifies there the two solutions.


If you would like to review the solution to problem 3.5c, click on Solution

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