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Solution - Absolute value equations

Exact form:

x = - \frac{1}{8}, \frac{19}{4}

x=-\frac{1}{8} , \frac{19}{4}

Mixed number form:

x = - \frac{1}{8}, 4 \frac{3}{4}

x=-\frac{1}{8} , 4\frac{3}{4}

Decimal form:

x = - 0.125, 4.75

x=-0.125 , 4.75

See steps

Step-by-step explanation

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

$3 | 2 x - 3 | + 2 | x + 5 | = 0$

Add $- 2 | x + 5 |$ to both sides of the equation:

$3 | 2 x - 3 | + 2 | x + 5 | - 2 | x + 5 | = - 2 | x + 5 |$

Simplify the arithmetic

$3 | 2 x - 3 | = - 2 | x + 5 |$

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
$3 | 2 x - 3 | = - 2 | x + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| 2 x - 3 \| = - 2 \| x + 5 \|$
$x = + y$	$3 (2 x - 3) = - 2 (x + 5)$
$x = - y$	$3 (2 x - 3) = - 2 (- (x + 5))$
$+ x = y$	$3 (2 x - 3) = - 2 (x + 5)$
$- x = y$	$3 (- (2 x - 3)) = - 2 (x + 5)$

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:

$\| x \| = \| y \|$	$3 \| 2 x - 3 \| = - 2 \| x + 5 \|$
$x = + y$ , $+ x = y$	$3 (2 x - 3) = - 2 (x + 5)$
$x = - y$ , $- x = y$	$3 (2 x - 3) = - 2 (- (x + 5))$

3. Solve the two equations for x

14 additional steps

$3 \cdot (2 x - 3) = - 2 \cdot (x + 5)$

Expand the parentheses:

$3 \cdot 2 x + 3 \cdot - 3 = - 2 \cdot (x + 5)$

Multiply the coefficients:

$6 x + 3 \cdot - 3 = - 2 \cdot (x + 5)$

Simplify the arithmetic:

$6 x - 9 = - 2 \cdot (x + 5)$

Expand the parentheses:

$6 x - 9 = - 2 x - 2 \cdot 5$

Simplify the arithmetic:

$6 x - 9 = - 2 x - 10$

Add to both sides:

$(6 x - 9) + 2 x = (- 2 x - 10) + 2 x$

Group like terms:

$(6 x + 2 x) - 9 = (- 2 x - 10) + 2 x$

Simplify the arithmetic:

$8 x - 9 = (- 2 x - 10) + 2 x$

Group like terms:

$8 x - 9 = (- 2 x + 2 x) - 10$

Simplify the arithmetic:

$8 x - 9 = - 10$

Add to both sides:

$(8 x - 9) + 9 = - 10 + 9$

Simplify the arithmetic:

$8 x = - 10 + 9$

Simplify the arithmetic:

$8 x = - 1$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{- 1}{8}$

Simplify the fraction:

$x = \frac{- 1}{8}$

17 additional steps

$3 \cdot (2 x - 3) = - 2 \cdot (- (x + 5))$

Expand the parentheses:

$3 \cdot 2 x + 3 \cdot - 3 = - 2 \cdot (- (x + 5))$

Multiply the coefficients:

$6 x + 3 \cdot - 3 = - 2 \cdot (- (x + 5))$

Simplify the arithmetic:

$6 x - 9 = - 2 \cdot (- (x + 5))$

Expand the parentheses:

$6 x - 9 = - 2 \cdot (- x - 5)$

$6 x - 9 = - 2 \cdot - x - 2 \cdot - 5$

Group like terms:

$6 x - 9 = (- 2 \cdot - 1) x - 2 \cdot - 5$

Multiply the coefficients:

$6 x - 9 = 2 x - 2 \cdot - 5$

Simplify the arithmetic:

$6 x - 9 = 2 x + 10$

Subtract from both sides:

$(6 x - 9) - 2 x = (2 x + 10) - 2 x$

Group like terms:

$(6 x - 2 x) - 9 = (2 x + 10) - 2 x$

Simplify the arithmetic:

$4 x - 9 = (2 x + 10) - 2 x$

Group like terms:

$4 x - 9 = (2 x - 2 x) + 10$

Simplify the arithmetic:

$4 x - 9 = 10$

Add to both sides:

$(4 x - 9) + 9 = 10 + 9$

Simplify the arithmetic:

$4 x = 10 + 9$

Simplify the arithmetic:

$4 x = 19$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{19}{4}$

Simplify the fraction:

$x = \frac{19}{4}$

4. List the solutions

$x = - \frac{1}{8}, \frac{19}{4}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 3 | 2 x - 3 |$
$y = - 2 | x + 5 |$
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