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Solution - Absolute value equations

Exact form:

x = - \frac{13}{2}, \frac{1}{4}

x=-\frac{13}{2} , \frac{1}{4}

Mixed number form:

x = - 6 \frac{1}{2}, \frac{1}{4}

x=-6\frac{1}{2} , \frac{1}{4}

Decimal form:

x = - 6.5, 0.25

x=-6.5 , 0.25

See steps

Step-by-step explanation

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

$| - x + 7 | + 3 | x + 2 | = 0$

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

$| - x + 7 | + 3 | x + 2 | - 3 | x + 2 | = - 3 | x + 2 |$

Simplify the arithmetic

$| - x + 7 | = - 3 | x + 2 |$

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
$| - x + 7 | = - 3 | x + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - x + 7 \| = - 3 \| x + 2 \|$
$x = + y$	$(- x + 7) = - 3 (x + 2)$
$x = - y$	$(- x + 7) = - 3 (- (x + 2))$
$+ x = y$	$(- x + 7) = - 3 (x + 2)$
$- x = y$	$- (- x + 7) = - 3 (x + 2)$

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:

$\| x \| = \| y \|$	$\| - x + 7 \| = - 3 \| x + 2 \|$
$x = + y$ , $+ x = y$	$(- x + 7) = - 3 (x + 2)$
$x = - y$ , $- x = y$	$(- x + 7) = - 3 (- (x + 2))$

3. Solve the two equations for x

11 additional steps

$(- x + 7) = - 3 \cdot (x + 2)$

Expand the parentheses:

$(- x + 7) = - 3 x - 3 \cdot 2$

Simplify the arithmetic:

$(- x + 7) = - 3 x - 6$

Add to both sides:

$(- x + 7) + 3 x = (- 3 x - 6) + 3 x$

Group like terms:

$(- x + 3 x) + 7 = (- 3 x - 6) + 3 x$

Simplify the arithmetic:

$2 x + 7 = (- 3 x - 6) + 3 x$

Group like terms:

$2 x + 7 = (- 3 x + 3 x) - 6$

Simplify the arithmetic:

$2 x + 7 = - 6$

Subtract from both sides:

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

Simplify the arithmetic:

$2 x = - 6 - 7$

Simplify the arithmetic:

$2 x = - 13$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{- 13}{2}$

Simplify the fraction:

$x = \frac{- 13}{2}$

16 additional steps

$(- x + 7) = - 3 \cdot (- (x + 2))$

Expand the parentheses:

$(- x + 7) = - 3 \cdot (- x - 2)$

$(- x + 7) = - 3 \cdot - x - 3 \cdot - 2$

Group like terms:

$(- x + 7) = (- 3 \cdot - 1) x - 3 \cdot - 2$

Multiply the coefficients:

$(- x + 7) = 3 x - 3 \cdot - 2$

Simplify the arithmetic:

$(- x + 7) = 3 x + 6$

Subtract from both sides:

$(- x + 7) - 3 x = (3 x + 6) - 3 x$

Group like terms:

$(- x - 3 x) + 7 = (3 x + 6) - 3 x$

Simplify the arithmetic:

$- 4 x + 7 = (3 x + 6) - 3 x$

Group like terms:

$- 4 x + 7 = (3 x - 3 x) + 6$

Simplify the arithmetic:

$- 4 x + 7 = 6$

Subtract from both sides:

$(- 4 x + 7) - 7 = 6 - 7$

Simplify the arithmetic:

$- 4 x = 6 - 7$

Simplify the arithmetic:

$- 4 x = - 1$

Divide both sides by :

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

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 1}{- 4}$

Simplify the fraction:

$x = \frac{- 1}{- 4}$

Cancel out the negatives:

$x = \frac{1}{4}$

4. List the solutions

$x = - \frac{13}{2}, \frac{1}{4}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - x + 7 |$
$y = - 3 | x + 2 |$
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