Teaching Notes 3.9: Solving Absolute Value Equations That
Have Two Solutions, One Solution, or No Solution
Many of your students may be proficient at solving absolute value equations. However, many of
these same students may become confused if, after they have isolated the absolute value
expression, they find that it is equal to a negative number or that it equals zero.- Explain that absolute value equations may have two solutions, one solution, or no solution.
Depending on the abilities of your students, you may find it helpful to review 3.8: ‘‘Solv-
ing Absolute Value Equations,’’ which focuses on absolute value equations that have two
solutions. - Ask your students to consider this example:| 3 x+ 1 |=−5. Note that the absolute value
expression is isolated and that it equals−5. Because the absolute value of any number or
expression must be greater than or equal to zero, this equation has no solution. Emphasize
that the absolute value expression must be isolated before determining whether or not there
is a solution. For example,| 3 x+ 1 |− 7 =−5 may appear to have no solution. Yet when
it is rewritten as| 3 x+ 1 |=2, the absolute value expression is isolated and there are two
solutions. - Now ask your students to consider this example:| 2 x+ 4 |=0. Because zero is neither posi-
tive nor negative, only one equation can be written: 2x+ 4 =0, thereforex=−2. There is
only one solution. - Review the information and examples on theworksheet with your students. Make sure that
your students understand the steps for solving equations. If necessary, provide additional
examples for review.
EXTRA HELP:
Isolate the absolute value expression before you determine the number of solutions.ANSWER KEY:
(1)x=18 andx=− 12 (2)x=− 5 (3)No solution (4)y= 0
(5)x=4andx=− 6 (6)y=2andy=− 5 (7)x=7andx= 3 (8)No solution
(9)x=3andx=− 1 (10)No solution (11)No solution (12)x=−2andx=− 14
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(Challenge)Equations may vary. One correct response is|x− 2 |=0.
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