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Solution - Absolute value equations

Exact form:

x = - \frac{1}{15}, \frac{7}{5}

x=-\frac{1}{15} , \frac{7}{5}

Mixed number form:

x = - \frac{1}{15}, 1 \frac{2}{5}

x=-\frac{1}{15} , 1\frac{2}{5}

Decimal form:

x = - 0.067, 1.4

x=-0.067 , 1.4

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
$| 5 x + 4 | = | - 10 x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x + 4 \| = \| - 10 x + 3 \|$
$x = + y$	$(5 x + 4) = (- 10 x + 3)$
$x = - y$	$(5 x + 4) = - (- 10 x + 3)$
$+ x = y$	$(5 x + 4) = (- 10 x + 3)$
$- x = y$	$- (5 x + 4) = (- 10 x + 3)$

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:

$\| x \| = \| y \|$	$\| 5 x + 4 \| = \| - 10 x + 3 \|$
$x = + y$ , $+ x = y$	$(5 x + 4) = (- 10 x + 3)$
$x = - y$ , $- x = y$	$(5 x + 4) = - (- 10 x + 3)$

2. Solve the two equations for x

9 additional steps

$(5 x + 4) = (- 10 x + 3)$

Add to both sides:

$(5 x + 4) + 10 x = (- 10 x + 3) + 10 x$

Group like terms:

$(5 x + 10 x) + 4 = (- 10 x + 3) + 10 x$

Simplify the arithmetic:

$15 x + 4 = (- 10 x + 3) + 10 x$

Group like terms:

$15 x + 4 = (- 10 x + 10 x) + 3$

Simplify the arithmetic:

$15 x + 4 = 3$

Subtract from both sides:

$(15 x + 4) - 4 = 3 - 4$

Simplify the arithmetic:

$15 x = 3 - 4$

Simplify the arithmetic:

$15 x = - 1$

Divide both sides by :

$\frac{(15 x)}{15} = \frac{- 1}{15}$

Simplify the fraction:

$x = \frac{- 1}{15}$

12 additional steps

$(5 x + 4) = - (- 10 x + 3)$

Expand the parentheses:

$(5 x + 4) = 10 x - 3$

Subtract from both sides:

$(5 x + 4) - 10 x = (10 x - 3) - 10 x$

Group like terms:

$(5 x - 10 x) + 4 = (10 x - 3) - 10 x$

Simplify the arithmetic:

$- 5 x + 4 = (10 x - 3) - 10 x$

Group like terms:

$- 5 x + 4 = (10 x - 10 x) - 3$

Simplify the arithmetic:

$- 5 x + 4 = - 3$

Subtract from both sides:

$(- 5 x + 4) - 4 = - 3 - 4$

Simplify the arithmetic:

$- 5 x = - 3 - 4$

Simplify the arithmetic:

$- 5 x = - 7$

Divide both sides by :

$\frac{(- 5 x)}{- 5} = \frac{- 7}{- 5}$

Cancel out the negatives:

$\frac{5 x}{5} = \frac{- 7}{- 5}$

Simplify the fraction:

$x = \frac{- 7}{- 5}$

Cancel out the negatives:

$x = \frac{7}{5}$

3. List the solutions

$x = - \frac{1}{15}, \frac{7}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 x + 4 |$
$y = | - 10 x + 3 |$
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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