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Solution - Absolute value equations

Exact form:

x = \frac{3}{2}, \frac{7}{3}

x=\frac{3}{2} , \frac{7}{3}

Mixed number form:

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

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

Decimal form:

x = 1.5, 2.333

x=1.5 , 2.333

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

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

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:

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

2. Solve the two equations for x

13 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 4 x - 4 = - 10$

Add to both sides:

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

Simplify the arithmetic:

$- 4 x = - 10 + 4$

Simplify the arithmetic:

$- 4 x = - 6$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 6}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 6}{- 4}$

Simplify the fraction:

$x = \frac{- 6}{- 4}$

Cancel out the negatives:

$x = \frac{6}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{3}{2}$

12 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$6 x - 4 = (- 5 x + 10) + 5 x$

Group like terms:

$6 x - 4 = (- 5 x + 5 x) + 10$

Simplify the arithmetic:

$6 x - 4 = 10$

Add to both sides:

$(6 x - 4) + 4 = 10 + 4$

Simplify the arithmetic:

$6 x = 10 + 4$

Simplify the arithmetic:

$6 x = 14$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{14}{6}$

Simplify the fraction:

$x = \frac{14}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(7 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{7}{3}$

3. List the solutions

$x = \frac{3}{2}, \frac{7}{3}$
(2 solution(s))

4. Graph

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