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Solution - Absolute value equations

Exact form:

x = - \frac{2}{61}, \frac{2}{63}

x=-\frac{2}{61} , \frac{2}{63}

Decimal form:

x = - 0.033, 0.032

x=-0.033 , 0.032

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| x - 2 | = | 62 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 2 \| = \| 62 x \|$
$x = + y$	$(x - 2) = (62 x)$
$x = - y$	$(x - 2) = - (62 x)$
$+ x = y$	$(x - 2) = (62 x)$
$- x = y$	$- (x - 2) = (62 x)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| x - 2 \| = \| 62 x \|$
$x = + y$ , $+ x = y$	$(x - 2) = (62 x)$
$x = - y$ , $- x = y$	$(x - 2) = - (62 x)$

2. Solve the two equations for x

10 additional steps

$(x - 2) = 62 x$

Subtract from both sides:

$(x - 2) - 62 x = (62 x) - 62 x$

Group like terms:

$(x - 62 x) - 2 = (62 x) - 62 x$

Simplify the arithmetic:

$- 61 x - 2 = (62 x) - 62 x$

Simplify the arithmetic:

$- 61 x - 2 = 0$

Add to both sides:

$(- 61 x - 2) + 2 = 0 + 2$

Simplify the arithmetic:

$- 61 x = 0 + 2$

Simplify the arithmetic:

$- 61 x = 2$

Divide both sides by :

$\frac{(- 61 x)}{- 61} = \frac{2}{- 61}$

Cancel out the negatives:

$\frac{61 x}{61} = \frac{2}{- 61}$

Simplify the fraction:

$x = \frac{2}{- 61}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 2}{61}$

7 additional steps

$(x - 2) = - 62 x$

Add to both sides:

$(x - 2) + 2 = (- 62 x) + 2$

Simplify the arithmetic:

$x = (- 62 x) + 2$

Add to both sides:

$x + 62 x = ((- 62 x) + 2) + 62 x$

Simplify the arithmetic:

$63 x = ((- 62 x) + 2) + 62 x$

Group like terms:

$63 x = (- 62 x + 62 x) + 2$

Simplify the arithmetic:

$63 x = 2$

Divide both sides by :

$\frac{(63 x)}{63} = \frac{2}{63}$

Simplify the fraction:

$x = \frac{2}{63}$

3. List the solutions

$x = - \frac{2}{61}, \frac{2}{63}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x - 2 |$
$y = | 62 x |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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