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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

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

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

Decimal form:

x = 1.4, 1

x=1.4 , 1

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
$| - x + 2 | = | 4 x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - x + 2 \| = \| 4 x - 5 \|$
$x = + y$	$(- x + 2) = (4 x - 5)$
$x = - y$	$(- x + 2) = - (4 x - 5)$
$+ x = y$	$(- x + 2) = (4 x - 5)$
$- x = y$	$- (- x + 2) = (4 x - 5)$

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:

$\| x \| = \| y \|$	$\| - x + 2 \| = \| 4 x - 5 \|$
$x = + y$ , $+ x = y$	$(- x + 2) = (4 x - 5)$
$x = - y$ , $- x = y$	$(- x + 2) = - (4 x - 5)$

2. Solve the two equations for x

11 additional steps

$(- x + 2) = (4 x - 5)$

Subtract from both sides:

$(- x + 2) - 4 x = (4 x - 5) - 4 x$

Group like terms:

$(- x - 4 x) + 2 = (4 x - 5) - 4 x$

Simplify the arithmetic:

$- 5 x + 2 = (4 x - 5) - 4 x$

Group like terms:

$- 5 x + 2 = (4 x - 4 x) - 5$

Simplify the arithmetic:

$- 5 x + 2 = - 5$

Subtract from both sides:

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

Simplify the arithmetic:

$- 5 x = - 5 - 2$

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:

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

11 additional steps

$(- x + 2) = - (4 x - 5)$

Expand the parentheses:

$(- x + 2) = - 4 x + 5$

Add to both sides:

$(- x + 2) + 4 x = (- 4 x + 5) + 4 x$

Group like terms:

$(- x + 4 x) + 2 = (- 4 x + 5) + 4 x$

Simplify the arithmetic:

$3 x + 2 = (- 4 x + 5) + 4 x$

Group like terms:

$3 x + 2 = (- 4 x + 4 x) + 5$

Simplify the arithmetic:

$3 x + 2 = 5$

Subtract from both sides:

$(3 x + 2) - 2 = 5 - 2$

Simplify the arithmetic:

$3 x = 5 - 2$

Simplify the arithmetic:

$3 x = 3$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$x = 1$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | - x + 2 |$
$y = | 4 x - 5 |$
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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