Absolute Value and Equations

21 Absolute Value and Equations

$\bullet\qquad |a| = b \quad (b>0) \qquad \Longleftrightarrow\qquad a= b \quad\hbox{ or }\quad a=-b$

ex.

$\qquad |x-5| = 3 \qquad \Longleftrightarrow\qquad x-5= 3 \quad\hbox{ or }\quad x-5=-3$

$\qquad \qquad\Longleftrightarrow\qquad x= 8 \quad\hbox{ or }\quad x=2$

$\bullet\qquad |a| = |b| \quad\quad \qquad \Longleftrightarrow\qquad a= b \quad\hbox{ or }\quad a=-b$

ex.

$\qquad |x-5| = |-x+3| \qquad \Longleftrightarrow\qquad x-5= -x+3 \quad\hbox{ or }\quad x-5=-(-x+3)$

$\qquad \qquad\Longleftrightarrow\qquad (2x= 8) \quad\hbox{ or }\quad (0. x=2 \quad\hbox{impossible})$

$\qquad \qquad\Longleftrightarrow\qquad x=4$

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Exercise 1

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The possible solutions are : $\qquad$ $\displaystyle{x= -33 \qquad \hbox{or} \qquad x= 31}$ .

Exercise 2

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The possible solutions are : $\qquad$ $\displaystyle{x= - \frac{{9}}{{26}} \qquad \hbox{or} \qquad x= - \frac{{27}}{{16}} }$ .

Exercise 3

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The possible solutions are : $\qquad$ $\displaystyle{x= \frac{{2}}{{28}} \qquad \hbox{or} \qquad x= \frac{{14}}{{28}} }$ .

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