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Solution - Absolute value equations

Exact form:

x = \frac{5}{2}, - \frac{2}{7}

x=\frac{5}{2} , -\frac{2}{7}

Mixed number form:

x = 2 \frac{1}{2}, - \frac{2}{7}

x=2\frac{1}{2} , -\frac{2}{7}

Decimal form:

x = 2.5, - 0.286

x=2.5 , -0.286

See steps

Step-by-step explanation

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

$| 5 x + 7 | - | 9 x - 3 | = 0$

Add $| 9 x - 3 |$ to both sides of the equation:

$| 5 x + 7 | - | 9 x - 3 | + | 9 x - 3 | = | 9 x - 3 |$

Simplify the arithmetic

$| 5 x + 7 | = | 9 x - 3 |$

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
$| 5 x + 7 | = | 9 x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x + 7 \| = \| 9 x - 3 \|$
$x = + y$	$(5 x + 7) = (9 x - 3)$
$x = - y$	$(5 x + 7) = (- (9 x - 3))$
$+ x = y$	$(5 x + 7) = (9 x - 3)$
$- x = y$	$- (5 x + 7) = (9 x - 3)$

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:

$\| x \| = \| y \|$	$\| 5 x + 7 \| = \| 9 x - 3 \|$
$x = + y$ , $+ x = y$	$(5 x + 7) = (9 x - 3)$
$x = - y$ , $- x = y$	$(5 x + 7) = (- (9 x - 3))$

3. Solve the two equations for x

13 additional steps

$(5 x + 7) = (9 x - 3)$

Subtract from both sides:

$(5 x + 7) - 9 x = (9 x - 3) - 9 x$

Group like terms:

$(5 x - 9 x) + 7 = (9 x - 3) - 9 x$

Simplify the arithmetic:

$- 4 x + 7 = (9 x - 3) - 9 x$

Group like terms:

$- 4 x + 7 = (9 x - 9 x) - 3$

Simplify the arithmetic:

$- 4 x + 7 = - 3$

Subtract from both sides:

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

Simplify the arithmetic:

$- 4 x = - 3 - 7$

Simplify the arithmetic:

$- 4 x = - 10$

Divide both sides by :

$\frac{(- 4 x)}{- 4} = \frac{- 10}{- 4}$

Cancel out the negatives:

$\frac{4 x}{4} = \frac{- 10}{- 4}$

Simplify the fraction:

$x = \frac{- 10}{- 4}$

Cancel out the negatives:

$x = \frac{10}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

12 additional steps

$(5 x + 7) = - (9 x - 3)$

Expand the parentheses:

$(5 x + 7) = - 9 x + 3$

Add to both sides:

$(5 x + 7) + 9 x = (- 9 x + 3) + 9 x$

Group like terms:

$(5 x + 9 x) + 7 = (- 9 x + 3) + 9 x$

Simplify the arithmetic:

$14 x + 7 = (- 9 x + 3) + 9 x$

Group like terms:

$14 x + 7 = (- 9 x + 9 x) + 3$

Simplify the arithmetic:

$14 x + 7 = 3$

Subtract from both sides:

$(14 x + 7) - 7 = 3 - 7$

Simplify the arithmetic:

$14 x = 3 - 7$

Simplify the arithmetic:

$14 x = - 4$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{- 4}{14}$

Simplify the fraction:

$x = \frac{- 4}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 2 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 2}{7}$

4. List the solutions

$x = \frac{5}{2}, - \frac{2}{7}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 5 x + 7 |$
$y = | 9 x - 3 |$
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