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Solution - Absolute value equations

Exact form:

x = 1, \frac{11}{5}

x=1 , \frac{11}{5}

Mixed number form:

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

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

Decimal form:

x = 1, 2.2

x=1 , 2.2

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
$| x - 4 | = | 4 x - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x - 4 \| = \| 4 x - 7 \|$
$x = + y$	$(x - 4) = (4 x - 7)$
$x = - y$	$(x - 4) = - (4 x - 7)$
$+ x = y$	$(x - 4) = (4 x - 7)$
$- x = y$	$- (x - 4) = (4 x - 7)$

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:

$\| x \| = \| y \|$	$\| x - 4 \| = \| 4 x - 7 \|$
$x = + y$ , $+ x = y$	$(x - 4) = (4 x - 7)$
$x = - y$ , $- x = y$	$(x - 4) = - (4 x - 7)$

2. Solve the two equations for x

12 additional steps

$(x - 4) = (4 x - 7)$

Subtract from both sides:

$(x - 4) - 4 x = (4 x - 7) - 4 x$

Group like terms:

$(x - 4 x) - 4 = (4 x - 7) - 4 x$

Simplify the arithmetic:

$- 3 x - 4 = (4 x - 7) - 4 x$

Group like terms:

$- 3 x - 4 = (4 x - 4 x) - 7$

Simplify the arithmetic:

$- 3 x - 4 = - 7$

Add to both sides:

$(- 3 x - 4) + 4 = - 7 + 4$

Simplify the arithmetic:

$- 3 x = - 7 + 4$

Simplify the arithmetic:

$- 3 x = - 3$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

Simplify the fraction:

$x = 1$

10 additional steps

$(x - 4) = - (4 x - 7)$

Expand the parentheses:

$(x - 4) = - 4 x + 7$

Add to both sides:

$(x - 4) + 4 x = (- 4 x + 7) + 4 x$

Group like terms:

$(x + 4 x) - 4 = (- 4 x + 7) + 4 x$

Simplify the arithmetic:

$5 x - 4 = (- 4 x + 7) + 4 x$

Group like terms:

$5 x - 4 = (- 4 x + 4 x) + 7$

Simplify the arithmetic:

$5 x - 4 = 7$

Add to both sides:

$(5 x - 4) + 4 = 7 + 4$

Simplify the arithmetic:

$5 x = 7 + 4$

Simplify the arithmetic:

$5 x = 11$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{11}{5}$

Simplify the fraction:

$x = \frac{11}{5}$

3. List the solutions

$x = 1, \frac{11}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | x - 4 |$
$y = | 4 x - 7 |$
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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