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Solution - Absolute value equations

Exact form:

x = 0.475, 0.183

x=0.475 , 0.183

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| x + \frac{2}{5} | + | - 5 x + 1.5 | = 0$

Add $- | - 5 x + 1.5 |$ to both sides of the equation:

$| x + \frac{2}{5} | + | - 5 x + 1.5 | - | - 5 x + 1.5 | = - | - 5 x + 1.5 |$

Simplify the arithmetic

$| x + \frac{2}{5} | = - | - 5 x + 1.5 |$

2. 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
$| x + \frac{2}{5} | = - | - 5 x + 1.5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x + \frac{2}{5} \| = - \| - 5 x + 1.5 \|$
$x = + y$	$(x + \frac{2}{5}) = - (- 5 x + 1.5)$
$x = - y$	$(x + \frac{2}{5}) = - - (- 5 x + 1.5)$
$+ x = y$	$(x + \frac{2}{5}) = - (- 5 x + 1.5)$
$- x = y$	$- (x + \frac{2}{5}) = - (- 5 x + 1.5)$

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:

$\| x \| = \| y \|$	$\| x + \frac{2}{5} \| = - \| - 5 x + 1.5 \|$
$x = + y$ , $+ x = y$	$(x + \frac{2}{5}) = - (- 5 x + 1.5)$
$x = - y$ , $- x = y$	$(x + \frac{2}{5}) = - - (- 5 x + 1.5)$

3. Solve the two equations for x

17 additional steps

$(x + \frac{2}{5}) = - (- 5 x + 1.5)$

Expand the parentheses:

$(x + \frac{2}{5}) = 5 x - 1.5$

Subtract from both sides:

$(x + \frac{2}{5}) - 5 x = (5 x - 1.5) - 5 x$

Group like terms:

$(x - 5 x) + \frac{2}{5} = (5 x - 1.5) - 5 x$

Simplify the arithmetic:

$- 4 x + \frac{2}{5} = (5 x - 1.5) - 5 x$

Group like terms:

$- 4 x + \frac{2}{5} = (5 x - 5 x) - 1.5$

Simplify the arithmetic:

$- 4 x + \frac{2}{5} = - 1.5$

Subtract from both sides:

$(- 4 x + \frac{2}{5}) - \frac{2}{5} = - 1.5 - \frac{2}{5}$

Combine the fractions:

$- 4 x + \frac{(2 - 2)}{5} = - 1.5 - \frac{2}{5}$

Combine the numerators:

$- 4 x + \frac{0}{5} = - 1.5 - \frac{2}{5}$

Reduce the zero numerator:

$- 4 x + 0 = - 1.5 - \frac{2}{5}$

Simplify the arithmetic:

$- 4 x = - 1.5 - \frac{2}{5}$

Divide fraction for addition:

$- 4 x = - 1.5 - 0.4$

Simplify the arithmetic:

$- 4 x = - 1.9$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 1.9}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 1.9}{- 4}$

Simplify the fraction:

$x = \frac{- 1.9}{- 4}$

Cancel out the negatives:

$x = \frac{1.9}{4}$

Simplify the arithmetic:

$x = 0.475$

15 additional steps

$(x + \frac{2}{5}) = - (- (- 5 x + 1.5))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(x + \frac{2}{5}) = - 5 x + 1.5$

Add to both sides:

$(x + \frac{2}{5}) + 5 x = (- 5 x + 1.5) + 5 x$

Group like terms:

$(x + 5 x) + \frac{2}{5} = (- 5 x + 1.5) + 5 x$

Simplify the arithmetic:

$6 x + \frac{2}{5} = (- 5 x + 1.5) + 5 x$

Group like terms:

$6 x + \frac{2}{5} = (- 5 x + 5 x) + 1.5$

Simplify the arithmetic:

$6 x + \frac{2}{5} = 1.5$

Subtract from both sides:

$(6 x + \frac{2}{5}) - \frac{2}{5} = 1.5 - \frac{2}{5}$

Combine the fractions:

$6 x + \frac{(2 - 2)}{5} = 1.5 - \frac{2}{5}$

Combine the numerators:

$6 x + \frac{0}{5} = 1.5 - \frac{2}{5}$

Reduce the zero numerator:

$6 x + 0 = 1.5 - \frac{2}{5}$

Simplify the arithmetic:

$6 x = 1.5 - \frac{2}{5}$

Divide fraction for addition:

$6 x = 1.5 - 0.4$

Simplify the arithmetic:

$6 x = 1.1$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{1.1}{6}$

Simplify the fraction:

$x = \frac{1.1}{6}$

Simplify the arithmetic:

$x = 0.1833$

4. List the solutions

$x = 0.475, 0.183$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | x + \frac{2}{5} |$
$y = - | - 5 x + 1.5 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved