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Solution - Absolute value equations

Exact form:

x = \frac{1}{2}, \frac{5}{4}

x=\frac{1}{2} , \frac{5}{4}

Mixed number form:

x = \frac{1}{2}, 1 \frac{1}{4}

x=\frac{1}{2} , 1\frac{1}{4}

Decimal form:

x = 0.5, 1.25

x=0.5 , 1.25

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
$| - 7 x + 8 | = | - 5 x + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 7 x + 8 \| = \| - 5 x + 7 \|$
$x = + y$	$(- 7 x + 8) = (- 5 x + 7)$
$x = - y$	$(- 7 x + 8) = - (- 5 x + 7)$
$+ x = y$	$(- 7 x + 8) = (- 5 x + 7)$
$- x = y$	$- (- 7 x + 8) = (- 5 x + 7)$

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:

$\| x \| = \| y \|$	$\| - 7 x + 8 \| = \| - 5 x + 7 \|$
$x = + y$ , $+ x = y$	$(- 7 x + 8) = (- 5 x + 7)$
$x = - y$ , $- x = y$	$(- 7 x + 8) = - (- 5 x + 7)$

2. Solve the two equations for x

11 additional steps

$(- 7 x + 8) = (- 5 x + 7)$

Add to both sides:

$(- 7 x + 8) + 5 x = (- 5 x + 7) + 5 x$

Group like terms:

$(- 7 x + 5 x) + 8 = (- 5 x + 7) + 5 x$

Simplify the arithmetic:

$- 2 x + 8 = (- 5 x + 7) + 5 x$

Group like terms:

$- 2 x + 8 = (- 5 x + 5 x) + 7$

Simplify the arithmetic:

$- 2 x + 8 = 7$

Subtract from both sides:

$(- 2 x + 8) - 8 = 7 - 8$

Simplify the arithmetic:

$- 2 x = 7 - 8$

Simplify the arithmetic:

$- 2 x = - 1$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{- 1}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{- 1}{- 2}$

Simplify the fraction:

$x = \frac{- 1}{- 2}$

Cancel out the negatives:

$x = \frac{1}{2}$

14 additional steps

$(- 7 x + 8) = - (- 5 x + 7)$

Expand the parentheses:

$(- 7 x + 8) = 5 x - 7$

Subtract from both sides:

$(- 7 x + 8) - 5 x = (5 x - 7) - 5 x$

Group like terms:

$(- 7 x - 5 x) + 8 = (5 x - 7) - 5 x$

Simplify the arithmetic:

$- 12 x + 8 = (5 x - 7) - 5 x$

Group like terms:

$- 12 x + 8 = (5 x - 5 x) - 7$

Simplify the arithmetic:

$- 12 x + 8 = - 7$

Subtract from both sides:

$(- 12 x + 8) - 8 = - 7 - 8$

Simplify the arithmetic:

$- 12 x = - 7 - 8$

Simplify the arithmetic:

$- 12 x = - 15$

Divide both sides by :

$\frac{(- 12 x)}{- 12} = \frac{- 15}{- 12}$

Cancel out the negatives:

$\frac{12 x}{12} = \frac{- 15}{- 12}$

Simplify the fraction:

$x = \frac{- 15}{- 12}$

Cancel out the negatives:

$x = \frac{15}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(5 \cdot 3)}{(4 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{5}{4}$

3. List the solutions

$x = \frac{1}{2}, \frac{5}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 7 x + 8 |$
$y = | - 5 x + 7 |$
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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