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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

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

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

Decimal form:

x = - 0.5, - 2.5

x=-0.5 , -2.5

See steps

Step-by-step explanation

1. 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
$4 | x + 2 | = | 2 x + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$4 \| x + 2 \| = \| 2 x + 7 \|$
$x = + y$	$4 (x + 2) = (2 x + 7)$
$x = - y$	$4 (x + 2) = - (2 x + 7)$
$+ x = y$	$4 (x + 2) = (2 x + 7)$
$- x = y$	$4 (- (x + 2)) = (2 x + 7)$

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:

$\| x \| = \| y \|$	$4 \| x + 2 \| = \| 2 x + 7 \|$
$x = + y$ , $+ x = y$	$4 (x + 2) = (2 x + 7)$
$x = - y$ , $- x = y$	$4 (x + 2) = - (2 x + 7)$

2. Solve the two equations for x

11 additional steps

$4 \cdot (x + 2) = (2 x + 7)$

Expand the parentheses:

$4 x + 4 \cdot 2 = (2 x + 7)$

Simplify the arithmetic:

$4 x + 8 = (2 x + 7)$

Subtract from both sides:

$(4 x + 8) - 2 x = (2 x + 7) - 2 x$

Group like terms:

$(4 x - 2 x) + 8 = (2 x + 7) - 2 x$

Simplify the arithmetic:

$2 x + 8 = (2 x + 7) - 2 x$

Group like terms:

$2 x + 8 = (2 x - 2 x) + 7$

Simplify the arithmetic:

$2 x + 8 = 7$

Subtract from both sides:

$(2 x + 8) - 8 = 7 - 8$

Simplify the arithmetic:

$2 x = 7 - 8$

Simplify the arithmetic:

$2 x = - 1$

Divide both sides by :

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

Simplify the fraction:

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

14 additional steps

$4 \cdot (x + 2) = - (2 x + 7)$

Expand the parentheses:

$4 x + 4 \cdot 2 = - (2 x + 7)$

Simplify the arithmetic:

$4 x + 8 = - (2 x + 7)$

Expand the parentheses:

$4 x + 8 = - 2 x - 7$

Add to both sides:

$(4 x + 8) + 2 x = (- 2 x - 7) + 2 x$

Group like terms:

$(4 x + 2 x) + 8 = (- 2 x - 7) + 2 x$

Simplify the arithmetic:

$6 x + 8 = (- 2 x - 7) + 2 x$

Group like terms:

$6 x + 8 = (- 2 x + 2 x) - 7$

Simplify the arithmetic:

$6 x + 8 = - 7$

Subtract from both sides:

$(6 x + 8) - 8 = - 7 - 8$

Simplify the arithmetic:

$6 x = - 7 - 8$

Simplify the arithmetic:

$6 x = - 15$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{- 15}{6}$

Simplify the fraction:

$x = \frac{- 15}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 3)}{(2 \cdot 3)}$

Factor out and cancel the greatest common factor:

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

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = 4 | x + 2 |$
$y = | 2 x + 7 |$
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 without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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