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Solution - Absolute value equations

Exact form:

x = - \frac{1}{6}, \frac{5}{2}

x=-\frac{1}{6} , \frac{5}{2}

Mixed number form:

x = - \frac{1}{6}, 2 \frac{1}{2}

x=-\frac{1}{6} , 2\frac{1}{2}

Decimal form:

x = - 0.167, 2.5

x=-0.167 , 2.5

See steps

Step-by-step explanation

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

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

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

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

Simplify the arithmetic

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

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

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

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:

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

3. Solve the two equations for x

13 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

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

Simplify the arithmetic:

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

Expand the parentheses:

$4 x - 2 = - 2 x - 3$

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$6 x - 2 = (- 2 x - 3) + 2 x$

Group like terms:

$6 x - 2 = (- 2 x + 2 x) - 3$

Simplify the arithmetic:

$6 x - 2 = - 3$

Add to both sides:

$(6 x - 2) + 2 = - 3 + 2$

Simplify the arithmetic:

$6 x = - 3 + 2$

Simplify the arithmetic:

$6 x = - 1$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{- 1}{6}$

Simplify the fraction:

$x = \frac{- 1}{6}$

13 additional steps

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

Expand the parentheses:

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

Multiply the coefficients:

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

Simplify the arithmetic:

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

Resolve the double minus:

$4 x - 2 = 2 x + 3$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$2 x - 2 = (2 x + 3) - 2 x$

Group like terms:

$2 x - 2 = (2 x - 2 x) + 3$

Simplify the arithmetic:

$2 x - 2 = 3$

Add to both sides:

$(2 x - 2) + 2 = 3 + 2$

Simplify the arithmetic:

$2 x = 3 + 2$

Simplify the arithmetic:

$2 x = 5$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{5}{2}$

Simplify the fraction:

$x = \frac{5}{2}$

4. List the solutions

$x = - \frac{1}{6}, \frac{5}{2}$
(2 solution(s))

5. Graph

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