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Solution - Absolute value equations

Exact form:

x = - \frac{5}{9}

x=-\frac{5}{9}

Decimal form:

x = - 0.556

x=-0.556

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

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

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:

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

2. Solve the two equations for x

5 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 7 = - 3$

The statement is false:

$- 7 = - 3$

The equation is false so it has no solution.

14 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 18 x - 7 = (9 x + 3) - 9 x$

Group like terms:

$- 18 x - 7 = (9 x - 9 x) + 3$

Simplify the arithmetic:

$- 18 x - 7 = 3$

Add to both sides:

$(- 18 x - 7) + 7 = 3 + 7$

Simplify the arithmetic:

$- 18 x = 3 + 7$

Simplify the arithmetic:

$- 18 x = 10$

Divide both sides by :

$\frac{(- 18 x)}{- 18} = \frac{10}{- 18}$

Cancel out the negatives:

$\frac{18 x}{18} = \frac{10}{- 18}$

Simplify the fraction:

$x = \frac{10}{- 18}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 10}{18}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 2)}{(9 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 5}{9}$

3. Graph

Each line represents the function of one side of the equation:
$y = | - 9 x - 7 |$
$y = | - 9 x - 3 |$
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 without absolute value bars

2. Solve the two equations for x

3. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved