No Solutions
Given:
[tex]-3|9x -7| = 2[/tex]
Rewriting the given equation:
[tex]-3|9x -7| = 2 \\ |9x -7| = -\frac{2}{3}[/tex]
We have to realize that the right side of the equation, [tex]|9x -7|[/tex], will always be positive no matter what real values of [tex]x[/tex] (because we're taking the absolute value of the expression) and we are equating it to a negative constant number, [tex]-\frac{2}{3}\\[/tex]. Something that is always positive will never be negative so there's no value for [tex]x[/tex] that satisfies the solution.
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[tex]\rule{6.5cm}{0.5pt}[/tex]
Solving by positive of the expression:
[tex]9x -7 = -\frac{2}{3} \\ 9x = -\frac{2}{3} +7 \\ 9x = -\frac{2}{3} +\frac{21}{3} \\ 9x = \frac{19}{3} \\ 9x \times \frac{1}{9} = \frac{19}{3} \times \frac{1}{9} \\ x = \frac{19}{27}[/tex]
Solving by the negative of the expression:
[tex]-(9x -7)= -\frac{2}{3} \\ 9x -7 = \frac{2}{3} \\ 9x = \frac{2}{3} +7 \\ 9x = \frac{2}{3} +\frac{21}{3} \\ 9x = \frac{23}{3} \\ 9x \times \frac{1}{9} = \frac{23}{3} \times \frac{1}{9} \\ x = \frac{23}{27}[/tex]
Checking: [tex]x = \frac{19}{27}\\[/tex]
[tex]-3|9(\frac{19}{27}) -7| \stackrel{?}{=} 2 \\ -3|\frac{19}{3} -7| \stackrel{?}{=} 2 \\ -3|\frac{19}{3} -\frac{21}{3}| \stackrel{?}{=} 2 \\ -3|-\frac{2}{3}| \stackrel{?}{=} \\ -3(\frac{2}{3}) \stackrel{?}{=} 2 \\ -2 \stackrel{?}{=} 2 \\ -2 \neq 2[/tex]
[tex]x = \frac{19}{27}\\[/tex] is an extraneous solution.
Checking: [tex]x = \frac{23}{27}\\[/tex]
[tex]-3|9(\frac{23}{27}) -7| \stackrel{?}{=} 2 \\ -3|\frac{23}{3} -7| \stackrel{?}{=} 2 \\ -3|\frac{23}{3} -\frac{21}{3}| \stackrel{?}{=} 2 \\ -3|\frac{2}{3}| \stackrel{?}{=} \\ -3(\frac{2}{3}) \stackrel{?}{=} 2 \\ -2 \stackrel{?}{=} 2 \\ -2 \neq 2[/tex]
[tex]x = \frac{23}{27}\\[/tex] is an extraneous solution.
