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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = - 0.4, - 0.4

x=-0.4 , -0.4

See steps

Step-by-step explanation

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

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

Add $- | x + \frac{2}{5} |$ to both sides of the equation:

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

Simplify the arithmetic

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

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

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

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:

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

3. Solve the two equations for x

14 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

$6 x = \frac{- 2}{5} + \frac{- 10}{5}$

Combine the fractions:

$6 x = \frac{(- 2 - 10)}{5}$

Combine the numerators:

$6 x = \frac{- 12}{5}$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{(\frac{- 12}{5})}{6}$

Simplify the fraction:

$x = \frac{(\frac{- 12}{5})}{6}$

Simplify the arithmetic:

$x = \frac{- 12}{(5 \cdot 6)}$

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

14 additional steps

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

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

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

Subtract from both sides:

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

Simplify the arithmetic:

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

Convert the integer into a fraction:

$4 x = \frac{2}{5} + \frac{- 10}{5}$

Combine the fractions:

$4 x = \frac{(2 - 10)}{5}$

Combine the numerators:

$4 x = \frac{- 8}{5}$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{(\frac{- 8}{5})}{4}$

Simplify the fraction:

$x = \frac{(\frac{- 8}{5})}{4}$

Simplify the arithmetic:

$x = \frac{- 8}{(5 \cdot 4)}$

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

4. List the solutions

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

5. Graph

Each line represents the function of one side of the equation:
$y = | 5 x + 2 |$
$y = - | x + \frac{2}{5} |$
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