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Solution - Absolute value equations

Exact form:

x = - \frac{23}{10}, - \frac{13}{8}

x=-\frac{23}{10} , -\frac{13}{8}

Mixed number form:

x = - 2 \frac{3}{10}, - 1 \frac{5}{8}

x=-2\frac{3}{10} , -1\frac{5}{8}

Decimal form:

x = - 2.3, - 1.625

x=-2.3 , -1.625

See steps

Step-by-step explanation

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

$| x + 5 | + | 9 x + 18 | = 0$

Add $- | 9 x + 18 |$ to both sides of the equation:

$| x + 5 | + | 9 x + 18 | - | 9 x + 18 | = - | 9 x + 18 |$

Simplify the arithmetic

$| x + 5 | = - | 9 x + 18 |$

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
$| x + 5 | = - | 9 x + 18 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x + 5 \| = - \| 9 x + 18 \|$
$x = + y$	$(x + 5) = - (9 x + 18)$
$x = - y$	$(x + 5) = - - (9 x + 18)$
$+ x = y$	$(x + 5) = - (9 x + 18)$
$- x = y$	$- (x + 5) = - (9 x + 18)$

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:

$\| x \| = \| y \|$	$\| x + 5 \| = - \| 9 x + 18 \|$
$x = + y$ , $+ x = y$	$(x + 5) = - (9 x + 18)$
$x = - y$ , $- x = y$	$(x + 5) = - - (9 x + 18)$

3. Solve the two equations for x

10 additional steps

$(x + 5) = - (9 x + 18)$

Expand the parentheses:

$(x + 5) = - 9 x - 18$

Add to both sides:

$(x + 5) + 9 x = (- 9 x - 18) + 9 x$

Group like terms:

$(x + 9 x) + 5 = (- 9 x - 18) + 9 x$

Simplify the arithmetic:

$10 x + 5 = (- 9 x - 18) + 9 x$

Group like terms:

$10 x + 5 = (- 9 x + 9 x) - 18$

Simplify the arithmetic:

$10 x + 5 = - 18$

Subtract from both sides:

$(10 x + 5) - 5 = - 18 - 5$

Simplify the arithmetic:

$10 x = - 18 - 5$

Simplify the arithmetic:

$10 x = - 23$

Divide both sides by :

$\frac{(10 x)}{10} = \frac{- 23}{10}$

Simplify the fraction:

$x = \frac{- 23}{10}$

12 additional steps

$(x + 5) = - (- (9 x + 18))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(x + 5) = 9 x + 18$

Subtract from both sides:

$(x + 5) - 9 x = (9 x + 18) - 9 x$

Group like terms:

$(x - 9 x) + 5 = (9 x + 18) - 9 x$

Simplify the arithmetic:

$- 8 x + 5 = (9 x + 18) - 9 x$

Group like terms:

$- 8 x + 5 = (9 x - 9 x) + 18$

Simplify the arithmetic:

$- 8 x + 5 = 18$

Subtract from both sides:

$(- 8 x + 5) - 5 = 18 - 5$

Simplify the arithmetic:

$- 8 x = 18 - 5$

Simplify the arithmetic:

$- 8 x = 13$

Divide both sides by :

$\frac{(- 8 x)}{- 8} = \frac{13}{- 8}$

Cancel out the negatives:

$\frac{8 x}{8} = \frac{13}{- 8}$

Simplify the fraction:

$x = \frac{13}{- 8}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 13}{8}$

4. List the solutions

$x = - \frac{23}{10}, - \frac{13}{8}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | x + 5 |$
$y = - | 9 x + 18 |$
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