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Solution - Absolute value equations

Exact form:

x = \frac{57}{4}, - \frac{15}{8}

x=\frac{57}{4} , -\frac{15}{8}

Mixed number form:

x = 14 \frac{1}{4}, - 1 \frac{7}{8}

x=14\frac{1}{4} , -1\frac{7}{8}

Decimal form:

x = 14.25, - 1.875

x=14.25 , -1.875

See steps

Step-by-step explanation

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

$2 | x + 18 | - 3 | 2 x - 7 | = 0$

Add $3 | 2 x - 7 |$ to both sides of the equation:

$2 | x + 18 | - 3 | 2 x - 7 | + 3 | 2 x - 7 | = 3 | 2 x - 7 |$

Simplify the arithmetic

$2 | x + 18 | = 3 | 2 x - 7 |$

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
$2 | x + 18 | = 3 | 2 x - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| x + 18 \| = 3 \| 2 x - 7 \|$
$x = + y$	$2 (x + 18) = 3 (2 x - 7)$
$x = - y$	$2 (x + 18) = 3 (- (2 x - 7))$
$+ x = y$	$2 (x + 18) = 3 (2 x - 7)$
$- x = y$	$2 (- (x + 18)) = 3 (2 x - 7)$

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:

$\| x \| = \| y \|$	$2 \| x + 18 \| = 3 \| 2 x - 7 \|$
$x = + y$ , $+ x = y$	$2 (x + 18) = 3 (2 x - 7)$
$x = - y$ , $- x = y$	$2 (x + 18) = 3 (- (2 x - 7))$

3. Solve the two equations for x

16 additional steps

$2 \cdot (x + 18) = 3 \cdot (2 x - 7)$

Expand the parentheses:

$2 x + 2 \cdot 18 = 3 \cdot (2 x - 7)$

Simplify the arithmetic:

$2 x + 36 = 3 \cdot (2 x - 7)$

Expand the parentheses:

$2 x + 36 = 3 \cdot 2 x + 3 \cdot - 7$

Multiply the coefficients:

$2 x + 36 = 6 x + 3 \cdot - 7$

Simplify the arithmetic:

$2 x + 36 = 6 x - 21$

Subtract from both sides:

$(2 x + 36) - 6 x = (6 x - 21) - 6 x$

Group like terms:

$(2 x - 6 x) + 36 = (6 x - 21) - 6 x$

Simplify the arithmetic:

$- 4 x + 36 = (6 x - 21) - 6 x$

Group like terms:

$- 4 x + 36 = (6 x - 6 x) - 21$

Simplify the arithmetic:

$- 4 x + 36 = - 21$

Subtract from both sides:

$(- 4 x + 36) - 36 = - 21 - 36$

Simplify the arithmetic:

$- 4 x = - 21 - 36$

Simplify the arithmetic:

$- 4 x = - 57$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 57}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 57}{- 4}$

Simplify the fraction:

$x = \frac{- 57}{- 4}$

Cancel out the negatives:

$x = \frac{57}{4}$

15 additional steps

$2 \cdot (x + 18) = 3 \cdot (- (2 x - 7))$

Expand the parentheses:

$2 x + 2 \cdot 18 = 3 \cdot (- (2 x - 7))$

Simplify the arithmetic:

$2 x + 36 = 3 \cdot (- (2 x - 7))$

Expand the parentheses:

$2 x + 36 = 3 \cdot (- 2 x + 7)$

Expand the parentheses:

$2 x + 36 = 3 \cdot - 2 x + 3 \cdot 7$

Multiply the coefficients:

$2 x + 36 = - 6 x + 3 \cdot 7$

Simplify the arithmetic:

$2 x + 36 = - 6 x + 21$

Add to both sides:

$(2 x + 36) + 6 x = (- 6 x + 21) + 6 x$

Group like terms:

$(2 x + 6 x) + 36 = (- 6 x + 21) + 6 x$

Simplify the arithmetic:

$8 x + 36 = (- 6 x + 21) + 6 x$

Group like terms:

$8 x + 36 = (- 6 x + 6 x) + 21$

Simplify the arithmetic:

$8 x + 36 = 21$

Subtract from both sides:

$(8 x + 36) - 36 = 21 - 36$

Simplify the arithmetic:

$8 x = 21 - 36$

Simplify the arithmetic:

$8 x = - 15$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{- 15}{8}$

Simplify the fraction:

$x = \frac{- 15}{8}$

4. List the solutions

$x = \frac{57}{4}, - \frac{15}{8}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 2 | x + 18 |$
$y = 3 | 2 x - 7 |$
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