Linear Sentences in One Variable
Segments, Lines, and Inequalities
Recall that the absolute value of a number represents the distance that number is from zero on the number line. The equation
is translated as “
x is 3 units from zero on the number line.” Notice, on the number line shown in Figure
1 , that there are two different numbers that are 3 units away from zero, namely, 3 or –3.
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The solution set of the equation
is
, because
and
Example 1: Solve for
This translates to “4
x – 2 is 8 units from zero on the number line” (see Figure
2 ).
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Check the solution.
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These are true statements. The solution set is
Example 2: Solve for
To solve this type of absolute value equation, first isolate the expression involving the absolute value symbol.
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Now, translate the absolute value equation:
is 11 units from zero on the number line.”
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The check is left to you. The solution set is
Example 3: Solve for
This problem has no solutions, because the translation is nonsensical. Distance is not measured in negative values.
Example 4: Solve for
This type of sentence will be true if either
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The expressions inside the absolute value symbols are exactly the same (that is, they are equal) or
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The expressions inside the absolute value symbols are opposites of each other.
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The check is left to you. The solution set is
Example 5: Solve for
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The sentence –2 = –7 is never true, so it gives no solution.
Check the solution.
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Therefore, the solution set is
