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Solution - Absolute value equations

Exact form:

x = \frac{7}{3}, 1

x=\frac{7}{3} , 1

Mixed number form:

x = 2 \frac{1}{3}, 1

x=2\frac{1}{3} , 1

Decimal form:

x = 2.333, 1

x=2.333 , 1

See steps

Step-by-step explanation

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

$2 | x + 1 | - 4 | 2 x - 3 | = 0$

Add $4 | 2 x - 3 |$ to both sides of the equation:

$2 | x + 1 | - 4 | 2 x - 3 | + 4 | 2 x - 3 | = 4 | 2 x - 3 |$

Simplify the arithmetic

$2 | x + 1 | = 4 | 2 x - 3 |$

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
$2 | x + 1 | = 4 | 2 x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| x + 1 \| = 4 \| 2 x - 3 \|$
$x = + y$	$2 (x + 1) = 4 (2 x - 3)$
$x = - y$	$2 (x + 1) = 4 (- (2 x - 3))$
$+ x = y$	$2 (x + 1) = 4 (2 x - 3)$
$- x = y$	$2 (- (x + 1)) = 4 (2 x - 3)$

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:

$\| x \| = \| y \|$	$2 \| x + 1 \| = 4 \| 2 x - 3 \|$
$x = + y$ , $+ x = y$	$2 (x + 1) = 4 (2 x - 3)$
$x = - y$ , $- x = y$	$2 (x + 1) = 4 (- (2 x - 3))$

3. Solve the two equations for x

18 additional steps

$2 \cdot (x + 1) = 4 \cdot (2 x - 3)$

Expand the parentheses:

$2 x + 2 \cdot 1 = 4 \cdot (2 x - 3)$

Simplify the arithmetic:

$2 x + 2 = 4 \cdot (2 x - 3)$

Expand the parentheses:

$2 x + 2 = 4 \cdot 2 x + 4 \cdot - 3$

Multiply the coefficients:

$2 x + 2 = 8 x + 4 \cdot - 3$

Simplify the arithmetic:

$2 x + 2 = 8 x - 12$

Subtract from both sides:

$(2 x + 2) - 8 x = (8 x - 12) - 8 x$

Group like terms:

$(2 x - 8 x) + 2 = (8 x - 12) - 8 x$

Simplify the arithmetic:

$- 6 x + 2 = (8 x - 12) - 8 x$

Group like terms:

$- 6 x + 2 = (8 x - 8 x) - 12$

Simplify the arithmetic:

$- 6 x + 2 = - 12$

Subtract from both sides:

$(- 6 x + 2) - 2 = - 12 - 2$

Simplify the arithmetic:

$- 6 x = - 12 - 2$

Simplify the arithmetic:

$- 6 x = - 14$

Divide both sides by :

$\frac{(- 6 x)}{- 6} = \frac{- 14}{- 6}$

Cancel out the negatives:

$\frac{6 x}{6} = \frac{- 14}{- 6}$

Simplify the fraction:

$x = \frac{- 14}{- 6}$

Cancel out the negatives:

$x = \frac{14}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(7 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{7}{3}$

16 additional steps

$2 \cdot (x + 1) = 4 \cdot (- (2 x - 3))$

Expand the parentheses:

$2 x + 2 \cdot 1 = 4 \cdot (- (2 x - 3))$

Simplify the arithmetic:

$2 x + 2 = 4 \cdot (- (2 x - 3))$

Expand the parentheses:

$2 x + 2 = 4 \cdot (- 2 x + 3)$

Expand the parentheses:

$2 x + 2 = 4 \cdot - 2 x + 4 \cdot 3$

Multiply the coefficients:

$2 x + 2 = - 8 x + 4 \cdot 3$

Simplify the arithmetic:

$2 x + 2 = - 8 x + 12$

Add to both sides:

$(2 x + 2) + 8 x = (- 8 x + 12) + 8 x$

Group like terms:

$(2 x + 8 x) + 2 = (- 8 x + 12) + 8 x$

Simplify the arithmetic:

$10 x + 2 = (- 8 x + 12) + 8 x$

Group like terms:

$10 x + 2 = (- 8 x + 8 x) + 12$

Simplify the arithmetic:

$10 x + 2 = 12$

Subtract from both sides:

$(10 x + 2) - 2 = 12 - 2$

Simplify the arithmetic:

$10 x = 12 - 2$

Simplify the arithmetic:

$10 x = 10$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$x = 1$

4. List the solutions

$x = \frac{7}{3}, 1$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 2 | x + 1 |$
$y = 4 | 2 x - 3 |$
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