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Solution - Absolute value equations

Exact form:

x = 1

x=1

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
$| - 7 x - 9 | = | - 7 x + 23 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 7 x - 9 \| = \| - 7 x + 23 \|$
$x = + y$	$(- 7 x - 9) = (- 7 x + 23)$
$x = - y$	$(- 7 x - 9) = - (- 7 x + 23)$
$+ x = y$	$(- 7 x - 9) = (- 7 x + 23)$
$- x = y$	$- (- 7 x - 9) = (- 7 x + 23)$

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:

$\| x \| = \| y \|$	$\| - 7 x - 9 \| = \| - 7 x + 23 \|$
$x = + y$ , $+ x = y$	$(- 7 x - 9) = (- 7 x + 23)$
$x = - y$ , $- x = y$	$(- 7 x - 9) = - (- 7 x + 23)$

2. Solve the two equations for x

5 additional steps

$(- 7 x - 9) = (- 7 x + 23)$

Add to both sides:

$(- 7 x - 9) + 7 x = (- 7 x + 23) + 7 x$

Group like terms:

$(- 7 x + 7 x) - 9 = (- 7 x + 23) + 7 x$

Simplify the arithmetic:

$- 9 = (- 7 x + 23) + 7 x$

Group like terms:

$- 9 = (- 7 x + 7 x) + 23$

Simplify the arithmetic:

$- 9 = 23$

The statement is false:

$- 9 = 23$

The equation is false so it has no solution.

13 additional steps

$(- 7 x - 9) = - (- 7 x + 23)$

Expand the parentheses:

$(- 7 x - 9) = 7 x - 23$

Subtract from both sides:

$(- 7 x - 9) - 7 x = (7 x - 23) - 7 x$

Group like terms:

$(- 7 x - 7 x) - 9 = (7 x - 23) - 7 x$

Simplify the arithmetic:

$- 14 x - 9 = (7 x - 23) - 7 x$

Group like terms:

$- 14 x - 9 = (7 x - 7 x) - 23$

Simplify the arithmetic:

$- 14 x - 9 = - 23$

Add to both sides:

$(- 14 x - 9) + 9 = - 23 + 9$

Simplify the arithmetic:

$- 14 x = - 23 + 9$

Simplify the arithmetic:

$- 14 x = - 14$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

Simplify the fraction:

$x = 1$

3. Graph

Each line represents the function of one side of the equation:
$y = | - 7 x - 9 |$
$y = | - 7 x + 23 |$
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