Solve the following absolute value equations: 6|1-5x|-9=57 and 5|9-5n|-7=38
Understand the Problem
The question asks to solve two absolute value equations. The first equation is 6|1-5x|-9=57 and the second is 5|9-5n|-7=38. We must isolate the absolute value, and then solve for both the positive, and negative values.
Answer
$x = -2, \frac{12}{5}$ $n = 0, \frac{18}{5}$
Answer for screen readers
$x = -2, \frac{12}{5}$ $n = 0, \frac{18}{5}$
Steps to Solve
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Isolate the absolute value for the first equation Add 9 to both sides of the equation $6|1-5x|-9=57$ $6|1-5x| = 57 + 9$ $6|1-5x| = 66$ Divide both sides by 6: $|1-5x| = \frac{66}{6}$ $|1-5x| = 11$
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Solve for the positive value for the first equation Remove the absolute value and solve for the positive value: $1 - 5x = 11$ Subtract 1 from both sides: $-5x = 10$ Divide both sides by -5: $x = -2$
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Solve for the negative value for the first equation Remove the absolute value and solve for the negative value: $1 - 5x = -11$ Subtract 1 from both sides: $-5x = -12$ Divide both sides by -5: $x = \frac{12}{5}$
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Isolate the absolute value for the second equation Add 7 to both sides of the equation $5|9-5n|-7=38$ $5|9-5n| = 38 + 7$ $5|9-5n| = 45$ Divide both sides by 5: $|9-5n| = \frac{45}{5}$ $|9-5n| = 9$
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Solve for the positive value for the second equation Remove the absolute value and solve for the positive value: $9 - 5n = 9$ Subtract 9 from both sides: $-5n = 0$ Divide both sides by -5: $n = 0$
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Solve for the negative value for the second equation Remove the absolute value and solve for the negative value: $9 - 5n = -9$ Subtract 9 from both sides: $-5n = -18$ Divide both sides by -5: $n = \frac{18}{5}$
$x = -2, \frac{12}{5}$ $n = 0, \frac{18}{5}$
More Information
Absolute value equations can have two solutions because the expression inside the absolute value can be either positive or negative and still result in the same absolute value.
Tips
A common mistake is forgetting to solve for both the positive and negative values after isolating the absolute value. Also, mistakes can be made during algebraic manipulation, such as not distributing a negative sign correctly or incorrectly adding or subtracting.
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