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Solution - Absolute value equations

Exact form:

x = \frac{9}{7}, \frac{9}{7}

x=\frac{9}{7} , \frac{9}{7}

Mixed number form:

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

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

Decimal form:

x = 1.286, 1.286

x=1.286 , 1.286

See steps

Step-by-step explanation

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

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

Add $- 5 | - 7 x + 9 |$ to both sides of the equation:

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

Simplify the arithmetic

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

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

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

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:

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

3. Solve the two equations for x

16 additional steps

$(7 x - 9) = - 5 \cdot (- 7 x + 9)$

Expand the parentheses:

$(7 x - 9) = - 5 \cdot - 7 x - 5 \cdot 9$

Multiply the coefficients:

$(7 x - 9) = 35 x - 5 \cdot 9$

Simplify the arithmetic:

$(7 x - 9) = 35 x - 45$

Subtract from both sides:

$(7 x - 9) - 35 x = (35 x - 45) - 35 x$

Group like terms:

$(7 x - 35 x) - 9 = (35 x - 45) - 35 x$

Simplify the arithmetic:

$- 28 x - 9 = (35 x - 45) - 35 x$

Group like terms:

$- 28 x - 9 = (35 x - 35 x) - 45$

Simplify the arithmetic:

$- 28 x - 9 = - 45$

Add to both sides:

$(- 28 x - 9) + 9 = - 45 + 9$

Simplify the arithmetic:

$- 28 x = - 45 + 9$

Simplify the arithmetic:

$- 28 x = - 36$

Divide both sides by :

$\frac{(- 28 x)}{- 28} = \frac{- 36}{- 28}$

Cancel out the negatives:

$\frac{28 x}{28} = \frac{- 36}{- 28}$

Simplify the fraction:

$x = \frac{- 36}{- 28}$

Cancel out the negatives:

$x = \frac{36}{28}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(9 \cdot 4)}{(7 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{9}{7}$

15 additional steps

$(7 x - 9) = - 5 \cdot (- (- 7 x + 9))$

Expand the parentheses:

$(7 x - 9) = - 5 \cdot (7 x - 9)$

Expand the parentheses:

$(7 x - 9) = - 5 \cdot 7 x - 5 \cdot - 9$

Multiply the coefficients:

$(7 x - 9) = - 35 x - 5 \cdot - 9$

Simplify the arithmetic:

$(7 x - 9) = - 35 x + 45$

Add to both sides:

$(7 x - 9) + 35 x = (- 35 x + 45) + 35 x$

Group like terms:

$(7 x + 35 x) - 9 = (- 35 x + 45) + 35 x$

Simplify the arithmetic:

$42 x - 9 = (- 35 x + 45) + 35 x$

Group like terms:

$42 x - 9 = (- 35 x + 35 x) + 45$

Simplify the arithmetic:

$42 x - 9 = 45$

Add to both sides:

$(42 x - 9) + 9 = 45 + 9$

Simplify the arithmetic:

$42 x = 45 + 9$

Simplify the arithmetic:

$42 x = 54$

Divide both sides by :

$\frac{(42 x)}{42} = \frac{54}{42}$

Simplify the fraction:

$x = \frac{54}{42}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(9 \cdot 6)}{(7 \cdot 6)}$

Factor out and cancel the greatest common factor:

$x = \frac{9}{7}$

4. List the solutions

$x = \frac{9}{7}, \frac{9}{7}$
(2 solution(s))

5. Graph

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