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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

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

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

Decimal form:

x = - 2.5, - 0.5

x=-2.5 , -0.5

See steps

Step-by-step explanation

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

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

Add $| 2 x - 1 |$ to both sides of the equation:

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

Simplify the arithmetic

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

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

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

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:

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

3. Solve the two equations for x

11 additional steps

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

Expand the parentheses:

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

Simplify the arithmetic:

$4 x + 4 = (2 x - 1)$

Subtract from both sides:

$(4 x + 4) - 2 x = (2 x - 1) - 2 x$

Group like terms:

$(4 x - 2 x) + 4 = (2 x - 1) - 2 x$

Simplify the arithmetic:

$2 x + 4 = (2 x - 1) - 2 x$

Group like terms:

$2 x + 4 = (2 x - 2 x) - 1$

Simplify the arithmetic:

$2 x + 4 = - 1$

Subtract from both sides:

$(2 x + 4) - 4 = - 1 - 4$

Simplify the arithmetic:

$2 x = - 1 - 4$

Simplify the arithmetic:

$2 x = - 5$

Divide both sides by :

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

Simplify the fraction:

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

14 additional steps

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

Expand the parentheses:

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

Simplify the arithmetic:

$4 x + 4 = (- (2 x - 1))$

Expand the parentheses:

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

Add to both sides:

$(4 x + 4) + 2 x = (- 2 x + 1) + 2 x$

Group like terms:

$(4 x + 2 x) + 4 = (- 2 x + 1) + 2 x$

Simplify the arithmetic:

$6 x + 4 = (- 2 x + 1) + 2 x$

Group like terms:

$6 x + 4 = (- 2 x + 2 x) + 1$

Simplify the arithmetic:

$6 x + 4 = 1$

Subtract from both sides:

$(6 x + 4) - 4 = 1 - 4$

Simplify the arithmetic:

$6 x = 1 - 4$

Simplify the arithmetic:

$6 x = - 3$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{- 3}{6}$

Simplify the fraction:

$x = \frac{- 3}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 3)}{(2 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{2}$

4. List the solutions

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

5. Graph

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