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Solution - Absolute value equations

Exact form:

z = \frac{20}{7}, \frac{4}{9}

z=\frac{20}{7} , \frac{4}{9}

Mixed number form:

z = 2 \frac{6}{7}, \frac{4}{9}

z=2\frac{6}{7} , \frac{4}{9}

Decimal form:

z = 2.857, 0.444

z=2.857 , 0.444

See steps

Step-by-step explanation

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

$| \frac{1}{2} z + 4 | - | 4 z - 6 | = 0$

Add $| 4 z - 6 |$ to both sides of the equation:

$| \frac{1}{2} z + 4 | - | 4 z - 6 | + | 4 z - 6 | = | 4 z - 6 |$

Simplify the arithmetic

$| \frac{1}{2} z + 4 | = | 4 z - 6 |$

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
$| \frac{1}{2} z + 4 | = | 4 z - 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| \frac{1}{2} z + 4 \| = \| 4 z - 6 \|$
$x = + y$	$(\frac{1}{2} z + 4) = (4 z - 6)$
$x = - y$	$(\frac{1}{2} z + 4) = (- (4 z - 6))$
$+ x = y$	$(\frac{1}{2} z + 4) = (4 z - 6)$
$- x = y$	$- (\frac{1}{2} z + 4) = (4 z - 6)$

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:

$\| x \| = \| y \|$	$\| \frac{1}{2} z + 4 \| = \| 4 z - 6 \|$
$x = + y$ , $+ x = y$	$(\frac{1}{2} z + 4) = (4 z - 6)$
$x = - y$ , $- x = y$	$(\frac{1}{2} z + 4) = (- (4 z - 6))$

3. Solve the two equations for z

19 additional steps

$(\frac{1}{2} z + 4) = (4 z - 6)$

Subtract from both sides:

$(\frac{1}{2} z + 4) - 4 z = (4 z - 6) - 4 z$

Group like terms:

$(\frac{1}{2} z - 4 z) + 4 = (4 z - 6) - 4 z$

Group the coefficients:

$(\frac{1}{2} - 4) z + 4 = (4 z - 6) - 4 z$

Convert the integer into a fraction:

$(\frac{1}{2} + \frac{- 8}{2}) z + 4 = (4 z - 6) - 4 z$

Combine the fractions:

$\frac{(1 - 8)}{2} z + 4 = (4 z - 6) - 4 z$

Combine the numerators:

$\frac{- 7}{2} z + 4 = (4 z - 6) - 4 z$

Group like terms:

$\frac{- 7}{2} z + 4 = (4 z - 4 z) - 6$

Simplify the arithmetic:

$\frac{- 7}{2} z + 4 = - 6$

Subtract from both sides:

$(\frac{- 7}{2} z + 4) - 4 = - 6 - 4$

Simplify the arithmetic:

$\frac{- 7}{2} z = - 6 - 4$

Simplify the arithmetic:

$\frac{- 7}{2} z = - 10$

Multiply both sides by inverse fraction :

$(\frac{- 7}{2} z) \cdot \frac{2}{- 7} = - 10 \cdot \frac{2}{- 7}$

Move the negative sign from the denominator to the numerator:

$\frac{- 7}{2} z \cdot \frac{- 2}{7} = - 10 \cdot \frac{2}{- 7}$

Group like terms:

$(\frac{- 7}{2} \cdot \frac{- 2}{7}) z = - 10 \cdot \frac{2}{- 7}$

Multiply the coefficients:

$\frac{(- 7 \cdot - 2)}{(2 \cdot 7)} z = - 10 \cdot \frac{2}{- 7}$

Simplify the arithmetic:

$1 z = - 10 \cdot \frac{2}{- 7}$

$z = - 10 \cdot \frac{2}{- 7}$

Move the negative sign from the denominator to the numerator:

$z = - 10 \cdot \frac{- 2}{7}$

Multiply the fraction(s):

$z = \frac{(- 10 \cdot - 2)}{7}$

Simplify the arithmetic:

$z = \frac{20}{7}$

17 additional steps

$(\frac{1}{2} z + 4) = - (4 z - 6)$

Expand the parentheses:

$(\frac{1}{2} z + 4) = - 4 z + 6$

Add to both sides:

$(\frac{1}{2} z + 4) + 4 z = (- 4 z + 6) + 4 z$

Group like terms:

$(\frac{1}{2} z + 4 z) + 4 = (- 4 z + 6) + 4 z$

Group the coefficients:

$(\frac{1}{2} + 4) z + 4 = (- 4 z + 6) + 4 z$

Convert the integer into a fraction:

$(\frac{1}{2} + \frac{8}{2}) z + 4 = (- 4 z + 6) + 4 z$

Combine the fractions:

$\frac{(1 + 8)}{2} z + 4 = (- 4 z + 6) + 4 z$

Combine the numerators:

$\frac{9}{2} z + 4 = (- 4 z + 6) + 4 z$

Group like terms:

$\frac{9}{2} z + 4 = (- 4 z + 4 z) + 6$

Simplify the arithmetic:

$\frac{9}{2} z + 4 = 6$

Subtract from both sides:

$(\frac{9}{2} z + 4) - 4 = 6 - 4$

Simplify the arithmetic:

$\frac{9}{2} z = 6 - 4$

Simplify the arithmetic:

$\frac{9}{2} z = 2$

Multiply both sides by inverse fraction :

$(\frac{9}{2} z) \cdot \frac{2}{9} = 2 \cdot \frac{2}{9}$

Group like terms:

$(\frac{9}{2} \cdot \frac{2}{9}) z = 2 \cdot \frac{2}{9}$

Multiply the coefficients:

$\frac{(9 \cdot 2)}{(2 \cdot 9)} z = 2 \cdot \frac{2}{9}$

Simplify the fraction:

$z = 2 \cdot \frac{2}{9}$

Multiply the fraction(s):

$z = \frac{(2 \cdot 2)}{9}$

Simplify the arithmetic:

$z = \frac{4}{9}$

4. List the solutions

$z = \frac{20}{7}, \frac{4}{9}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | \frac{1}{2} z + 4 |$
$y = | 4 z - 6 |$
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 z

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 z

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved