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Solution - Absolute value equations

Exact form:

x = \frac{7}{5}

x=\frac{7}{5}

Mixed number form:

x = 1 \frac{2}{5}

x=1\frac{2}{5}

Decimal form:

x = 1.4

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

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

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:

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

2. Solve the two equations for x

5 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 = 10$

The statement is false:

$4 = 10$

The equation is false so it has no solution.

14 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 10 x + 4 = - 10$

Subtract from both sides:

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

Simplify the arithmetic:

$- 10 x = - 10 - 4$

Simplify the arithmetic:

$- 10 x = - 14$

Divide both sides by :

$\frac{(- 10 x)}{- 10} = \frac{- 14}{- 10}$

Cancel out the negatives:

$\frac{10 x}{10} = \frac{- 14}{- 10}$

Simplify the fraction:

$x = \frac{- 14}{- 10}$

Cancel out the negatives:

$x = \frac{14}{10}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(7 \cdot 2)}{(5 \cdot 2)}$

Factor out and cancel the greatest common factor:

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

3. Graph

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

Why learn this

Terms and topics

Related links

Latest Related Drills Solved