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Solution - Absolute value equations

Exact form:

x = \frac{13}{6}, - \frac{5}{4}

x=\frac{13}{6} , -\frac{5}{4}

Mixed number form:

x = 2 \frac{1}{6}, - 1 \frac{1}{4}

x=2\frac{1}{6} , -1\frac{1}{4}

Decimal form:

x = 2.167, - 1.25

x=2.167 , -1.25

See steps

Step-by-step explanation

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

$| - 5 x + 4 | + | - x + 9 | = 0$

Add $- | - x + 9 |$ to both sides of the equation:

$| - 5 x + 4 | + | - x + 9 | - | - x + 9 | = - | - x + 9 |$

Simplify the arithmetic

$| - 5 x + 4 | = - | - x + 9 |$

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
$| - 5 x + 4 | = - | - x + 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 5 x + 4 \| = - \| - x + 9 \|$
$x = + y$	$(- 5 x + 4) = - (- x + 9)$
$x = - y$	$(- 5 x + 4) = - - (- x + 9)$
$+ x = y$	$(- 5 x + 4) = - (- x + 9)$
$- x = y$	$- (- 5 x + 4) = - (- x + 9)$

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:

$\| x \| = \| y \|$	$\| - 5 x + 4 \| = - \| - x + 9 \|$
$x = + y$ , $+ x = y$	$(- 5 x + 4) = - (- x + 9)$
$x = - y$ , $- x = y$	$(- 5 x + 4) = - - (- x + 9)$

3. Solve the two equations for x

12 additional steps

$(- 5 x + 4) = - (- x + 9)$

Expand the parentheses:

$(- 5 x + 4) = x - 9$

Subtract from both sides:

$(- 5 x + 4) - x = (x - 9) - x$

Group like terms:

$(- 5 x - x) + 4 = (x - 9) - x$

Simplify the arithmetic:

$- 6 x + 4 = (x - 9) - x$

Group like terms:

$- 6 x + 4 = (x - x) - 9$

Simplify the arithmetic:

$- 6 x + 4 = - 9$

Subtract from both sides:

$(- 6 x + 4) - 4 = - 9 - 4$

Simplify the arithmetic:

$- 6 x = - 9 - 4$

Simplify the arithmetic:

$- 6 x = - 13$

Divide both sides by :

$\frac{(- 6 x)}{- 6} = \frac{- 13}{- 6}$

Cancel out the negatives:

$\frac{6 x}{6} = \frac{- 13}{- 6}$

Simplify the fraction:

$x = \frac{- 13}{- 6}$

Cancel out the negatives:

$x = \frac{13}{6}$

12 additional steps

$(- 5 x + 4) = - (- (- x + 9))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(- 5 x + 4) = - x + 9$

Add to both sides:

$(- 5 x + 4) + x = (- x + 9) + x$

Group like terms:

$(- 5 x + x) + 4 = (- x + 9) + x$

Simplify the arithmetic:

$- 4 x + 4 = (- x + 9) + x$

Group like terms:

$- 4 x + 4 = (- x + x) + 9$

Simplify the arithmetic:

$- 4 x + 4 = 9$

Subtract from both sides:

$(- 4 x + 4) - 4 = 9 - 4$

Simplify the arithmetic:

$- 4 x = 9 - 4$

Simplify the arithmetic:

$- 4 x = 5$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

4. List the solutions

$x = \frac{13}{6}, - \frac{5}{4}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - 5 x + 4 |$
$y = - | - x + 9 |$
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