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Solution - Absolute value equations

Exact form:

x = 3, \frac{6}{11}

x=3 , \frac{6}{11}

Decimal form:

x = 3, 0.545

x=3 , 0.545

See steps

Step-by-step explanation

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

$| 7 x + 6 | - | 15 x - 18 | = 0$

Add $| 15 x - 18 |$ to both sides of the equation:

$| 7 x + 6 | - | 15 x - 18 | + | 15 x - 18 | = | 15 x - 18 |$

Simplify the arithmetic

$| 7 x + 6 | = | 15 x - 18 |$

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
$| 7 x + 6 | = | 15 x - 18 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 7 x + 6 \| = \| 15 x - 18 \|$
$x = + y$	$(7 x + 6) = (15 x - 18)$
$x = - y$	$(7 x + 6) = (- (15 x - 18))$
$+ x = y$	$(7 x + 6) = (15 x - 18)$
$- x = y$	$- (7 x + 6) = (15 x - 18)$

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:

$\| x \| = \| y \|$	$\| 7 x + 6 \| = \| 15 x - 18 \|$
$x = + y$ , $+ x = y$	$(7 x + 6) = (15 x - 18)$
$x = - y$ , $- x = y$	$(7 x + 6) = (- (15 x - 18))$

3. Solve the two equations for x

13 additional steps

$(7 x + 6) = (15 x - 18)$

Subtract from both sides:

$(7 x + 6) - 15 x = (15 x - 18) - 15 x$

Group like terms:

$(7 x - 15 x) + 6 = (15 x - 18) - 15 x$

Simplify the arithmetic:

$- 8 x + 6 = (15 x - 18) - 15 x$

Group like terms:

$- 8 x + 6 = (15 x - 15 x) - 18$

Simplify the arithmetic:

$- 8 x + 6 = - 18$

Subtract from both sides:

$(- 8 x + 6) - 6 = - 18 - 6$

Simplify the arithmetic:

$- 8 x = - 18 - 6$

Simplify the arithmetic:

$- 8 x = - 24$

Divide both sides by :

$\frac{(- 8 x)}{- 8} = \frac{- 24}{- 8}$

Cancel out the negatives:

$\frac{8 x}{8} = \frac{- 24}{- 8}$

Simplify the fraction:

$x = \frac{- 24}{- 8}$

Cancel out the negatives:

$x = \frac{24}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 8)}{(1 \cdot 8)}$

Factor out and cancel the greatest common factor:

$x = 3$

12 additional steps

$(7 x + 6) = - (15 x - 18)$

Expand the parentheses:

$(7 x + 6) = - 15 x + 18$

Add to both sides:

$(7 x + 6) + 15 x = (- 15 x + 18) + 15 x$

Group like terms:

$(7 x + 15 x) + 6 = (- 15 x + 18) + 15 x$

Simplify the arithmetic:

$22 x + 6 = (- 15 x + 18) + 15 x$

Group like terms:

$22 x + 6 = (- 15 x + 15 x) + 18$

Simplify the arithmetic:

$22 x + 6 = 18$

Subtract from both sides:

$(22 x + 6) - 6 = 18 - 6$

Simplify the arithmetic:

$22 x = 18 - 6$

Simplify the arithmetic:

$22 x = 12$

Divide both sides by :

$\frac{(22 x)}{22} = \frac{12}{22}$

Simplify the fraction:

$x = \frac{12}{22}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(6 \cdot 2)}{(11 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{6}{11}$

4. List the solutions

$x = 3, \frac{6}{11}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 7 x + 6 |$
$y = | 15 x - 18 |$
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