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Solution - Absolute value equations

Exact form:

x = - 4, - \frac{5}{2}

x=-4 , -\frac{5}{2}

Mixed number form:

x = - 4, - 2 \frac{1}{2}

x=-4 , -2\frac{1}{2}

Decimal form:

x = - 4, - 2.5

x=-4 , -2.5

See steps

Step-by-step explanation

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

$6 | x + 3 | - 2 | x + 1 | = 0$

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

$6 | x + 3 | - 2 | x + 1 | + 2 | x + 1 | = 2 | x + 1 |$

Simplify the arithmetic

$6 | x + 3 | = 2 | x + 1 |$

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
$6 | x + 3 | = 2 | x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$6 \| x + 3 \| = 2 \| x + 1 \|$
$x = + y$	$6 (x + 3) = 2 (x + 1)$
$x = - y$	$6 (x + 3) = 2 (- (x + 1))$
$+ x = y$	$6 (x + 3) = 2 (x + 1)$
$- x = y$	$6 (- (x + 3)) = 2 (x + 1)$

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:

$\| x \| = \| y \|$	$6 \| x + 3 \| = 2 \| x + 1 \|$
$x = + y$ , $+ x = y$	$6 (x + 3) = 2 (x + 1)$
$x = - y$ , $- x = y$	$6 (x + 3) = 2 (- (x + 1))$

3. Solve the two equations for x

15 additional steps

$6 \cdot (x + 3) = 2 \cdot (x + 1)$

Expand the parentheses:

$6 x + 6 \cdot 3 = 2 \cdot (x + 1)$

Simplify the arithmetic:

$6 x + 18 = 2 \cdot (x + 1)$

Expand the parentheses:

$6 x + 18 = 2 x + 2 \cdot 1$

Simplify the arithmetic:

$6 x + 18 = 2 x + 2$

Subtract from both sides:

$(6 x + 18) - 2 x = (2 x + 2) - 2 x$

Group like terms:

$(6 x - 2 x) + 18 = (2 x + 2) - 2 x$

Simplify the arithmetic:

$4 x + 18 = (2 x + 2) - 2 x$

Group like terms:

$4 x + 18 = (2 x - 2 x) + 2$

Simplify the arithmetic:

$4 x + 18 = 2$

Subtract from both sides:

$(4 x + 18) - 18 = 2 - 18$

Simplify the arithmetic:

$4 x = 2 - 18$

Simplify the arithmetic:

$4 x = - 16$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{- 16}{4}$

Simplify the fraction:

$x = \frac{- 16}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 4 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = - 4$

18 additional steps

$6 \cdot (x + 3) = 2 \cdot (- (x + 1))$

Expand the parentheses:

$6 x + 6 \cdot 3 = 2 \cdot (- (x + 1))$

Simplify the arithmetic:

$6 x + 18 = 2 \cdot (- (x + 1))$

Expand the parentheses:

$6 x + 18 = 2 \cdot (- x - 1)$

$6 x + 18 = 2 \cdot - x + 2 \cdot - 1$

Group like terms:

$6 x + 18 = (2 \cdot - 1) x + 2 \cdot - 1$

Multiply the coefficients:

$6 x + 18 = - 2 x + 2 \cdot - 1$

Simplify the arithmetic:

$6 x + 18 = - 2 x - 2$

Add to both sides:

$(6 x + 18) + 2 x = (- 2 x - 2) + 2 x$

Group like terms:

$(6 x + 2 x) + 18 = (- 2 x - 2) + 2 x$

Simplify the arithmetic:

$8 x + 18 = (- 2 x - 2) + 2 x$

Group like terms:

$8 x + 18 = (- 2 x + 2 x) - 2$

Simplify the arithmetic:

$8 x + 18 = - 2$

Subtract from both sides:

$(8 x + 18) - 18 = - 2 - 18$

Simplify the arithmetic:

$8 x = - 2 - 18$

Simplify the arithmetic:

$8 x = - 20$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{- 20}{8}$

Simplify the fraction:

$x = \frac{- 20}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 4)}{(2 \cdot 4)}$

Factor out and cancel the greatest common factor:

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

4. List the solutions

$x = - 4, - \frac{5}{2}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 6 | x + 3 |$
$y = 2 | x + 1 |$
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