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Solution - Absolute value equations

Exact form:

x = \frac{5}{3}, 15

x=\frac{5}{3} , 15

Mixed number form:

x = 1 \frac{2}{3}, 15

x=1\frac{2}{3} , 15

Decimal form:

x = 1.667, 15

x=1.667 , 15

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
$| - 2 x + 10 | = | x + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 x + 10 \| = \| x + 5 \|$
$x = + y$	$(- 2 x + 10) = (x + 5)$
$x = - y$	$(- 2 x + 10) = - (x + 5)$
$+ x = y$	$(- 2 x + 10) = (x + 5)$
$- x = y$	$- (- 2 x + 10) = (x + 5)$

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:

$\| x \| = \| y \|$	$\| - 2 x + 10 \| = \| x + 5 \|$
$x = + y$ , $+ x = y$	$(- 2 x + 10) = (x + 5)$
$x = - y$ , $- x = y$	$(- 2 x + 10) = - (x + 5)$

2. Solve the two equations for x

11 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 3 x + 10 = (x + 5) - x$

Group like terms:

$- 3 x + 10 = (x - x) + 5$

Simplify the arithmetic:

$- 3 x + 10 = 5$

Subtract from both sides:

$(- 3 x + 10) - 10 = 5 - 10$

Simplify the arithmetic:

$- 3 x = 5 - 10$

Simplify the arithmetic:

$- 3 x = - 5$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{- 5}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{- 5}{- 3}$

Simplify the fraction:

$x = \frac{- 5}{- 3}$

Cancel out the negatives:

$x = \frac{5}{3}$

11 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- x + 10 = - 5$

Subtract from both sides:

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

Simplify the arithmetic:

$- x = - 5 - 10$

Simplify the arithmetic:

$- x = - 15$

Multiply both sides by :

$- x \cdot - 1 = - 15 \cdot - 1$

Remove the one(s):

$x = - 15 \cdot - 1$

Simplify the arithmetic:

$x = 15$

3. List the solutions

$x = \frac{5}{3}, 15$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 2 x + 10 |$
$y = | x + 5 |$
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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