Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

z = - \frac{4}{7}, - 4

z=-\frac{4}{7} , -4

Decimal form:

z = - 0.571, - 4

z=-0.571 , -4

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
$| - 7 z - 4 | = | 7 z + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 7 z - 4 \| = \| 7 z + 4 \|$
$x = + y$	$(- 7 z - 4) = (7 z + 4)$
$x = - y$	$(- 7 z - 4) = - (7 z + 4)$
$+ x = y$	$(- 7 z - 4) = (7 z + 4)$
$- x = y$	$- (- 7 z - 4) = (7 z + 4)$

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:

$\| x \| = \| y \|$	$\| - 7 z - 4 \| = \| 7 z + 4 \|$
$x = + y$ , $+ x = y$	$(- 7 z - 4) = (7 z + 4)$
$x = - y$ , $- x = y$	$(- 7 z - 4) = - (7 z + 4)$

2. Solve the two equations for z

13 additional steps

$(- 7 z - 4) = (7 z + 4)$

Subtract from both sides:

$(- 7 z - 4) - 7 z = (7 z + 4) - 7 z$

Group like terms:

$(- 7 z - 7 z) - 4 = (7 z + 4) - 7 z$

Simplify the arithmetic:

$- 14 z - 4 = (7 z + 4) - 7 z$

Group like terms:

$- 14 z - 4 = (7 z - 7 z) + 4$

Simplify the arithmetic:

$- 14 z - 4 = 4$

Add to both sides:

$(- 14 z - 4) + 4 = 4 + 4$

Simplify the arithmetic:

$- 14 z = 4 + 4$

Simplify the arithmetic:

$- 14 z = 8$

Divide both sides by :

$\frac{(- 14 z)}{- 14} = \frac{8}{- 14}$

Cancel out the negatives:

$\frac{14 z}{14} = \frac{8}{- 14}$

Simplify the fraction:

$z = \frac{8}{- 14}$

Move the negative sign from the denominator to the numerator:

$z = \frac{- 8}{14}$

Find the greatest common factor of the numerator and denominator:

$z = \frac{(- 4 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$z = \frac{- 4}{7}$

5 additional steps

$(- 7 z - 4) = - (7 z + 4)$

Expand the parentheses:

$(- 7 z - 4) = - 7 z - 4$

Add to both sides:

$(- 7 z - 4) + 7 z = (- 7 z - 4) + 7 z$

Group like terms:

$(- 7 z + 7 z) - 4 = (- 7 z - 4) + 7 z$

Simplify the arithmetic:

$- 4 = (- 7 z - 4) + 7 z$

Group like terms:

$- 4 = (- 7 z + 7 z) - 4$

Simplify the arithmetic:

$- 4 = - 4$

3. List the solutions

$z = - \frac{4}{7}, - 4$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 7 z - 4 |$
$y = | 7 z + 4 |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

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 z

3. List the solutions

4. 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 z

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved