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Solution - Absolute value equations

Exact form:

x = \frac{8}{3}, \frac{12}{5}

x=\frac{8}{3} , \frac{12}{5}

Mixed number form:

x = 2 \frac{2}{3}, 2 \frac{2}{5}

x=2\frac{2}{3} , 2\frac{2}{5}

Decimal form:

x = 2.667, 2.4

x=2.667 , 2.4

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
$| 4 x - 10 | = | x - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 10 \| = \| x - 2 \|$
$x = + y$	$(4 x - 10) = (x - 2)$
$x = - y$	$(4 x - 10) = - (x - 2)$
$+ x = y$	$(4 x - 10) = (x - 2)$
$- x = y$	$- (4 x - 10) = (x - 2)$

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:

$\| x \| = \| y \|$	$\| 4 x - 10 \| = \| x - 2 \|$
$x = + y$ , $+ x = y$	$(4 x - 10) = (x - 2)$
$x = - y$ , $- x = y$	$(4 x - 10) = - (x - 2)$

2. Solve the two equations for x

9 additional steps

$(4 x - 10) = (x - 2)$

Subtract from both sides:

$(4 x - 10) - x = (x - 2) - x$

Group like terms:

$(4 x - x) - 10 = (x - 2) - x$

Simplify the arithmetic:

$3 x - 10 = (x - 2) - x$

Group like terms:

$3 x - 10 = (x - x) - 2$

Simplify the arithmetic:

$3 x - 10 = - 2$

Add to both sides:

$(3 x - 10) + 10 = - 2 + 10$

Simplify the arithmetic:

$3 x = - 2 + 10$

Simplify the arithmetic:

$3 x = 8$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{8}{3}$

Simplify the fraction:

$x = \frac{8}{3}$

10 additional steps

$(4 x - 10) = - (x - 2)$

Expand the parentheses:

$(4 x - 10) = - x + 2$

Add to both sides:

$(4 x - 10) + x = (- x + 2) + x$

Group like terms:

$(4 x + x) - 10 = (- x + 2) + x$

Simplify the arithmetic:

$5 x - 10 = (- x + 2) + x$

Group like terms:

$5 x - 10 = (- x + x) + 2$

Simplify the arithmetic:

$5 x - 10 = 2$

Add to both sides:

$(5 x - 10) + 10 = 2 + 10$

Simplify the arithmetic:

$5 x = 2 + 10$

Simplify the arithmetic:

$5 x = 12$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{12}{5}$

Simplify the fraction:

$x = \frac{12}{5}$

3. List the solutions

$x = \frac{8}{3}, \frac{12}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 10 |$
$y = | x - 2 |$
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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