Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

k = - \frac{20}{9}, \frac{20}{19}

k=-\frac{20}{9} , \frac{20}{19}

Mixed number form:

k = - 2 \frac{2}{9}, 1 \frac{1}{19}

k=-2\frac{2}{9} , 1\frac{1}{19}

Decimal form:

k = - 2.222, 1.053

k=-2.222 , 1.053

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
$| \frac{7}{5} k | = | \frac{1}{2} k - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| \frac{7}{5} k \| = \| \frac{1}{2} k - 2 \|$
$x = + y$	$(\frac{7}{5} k) = (\frac{1}{2} k - 2)$
$x = - y$	$(\frac{7}{5} k) = - (\frac{1}{2} k - 2)$
$+ x = y$	$(\frac{7}{5} k) = (\frac{1}{2} k - 2)$
$- x = y$	$- (\frac{7}{5} k) = (\frac{1}{2} k - 2)$

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:

$\| x \| = \| y \|$	$\| \frac{7}{5} k \| = \| \frac{1}{2} k - 2 \|$
$x = + y$ , $+ x = y$	$(\frac{7}{5} k) = (\frac{1}{2} k - 2)$
$x = - y$ , $- x = y$	$(\frac{7}{5} k) = - (\frac{1}{2} k - 2)$

2. Solve the two equations for k

17 additional steps

$\frac{7}{5} \cdot k = (\frac{1}{2} k - 2)$

Subtract from both sides:

$(\frac{7}{5} k) - \frac{1}{2} \cdot k = (\frac{1}{2} k - 2) - \frac{1}{2} k$

Group the coefficients:

$(\frac{7}{5} + \frac{- 1}{2}) k = (\frac{1}{2} \cdot k - 2) - \frac{1}{2} k$

Find the lowest common denominator:

$(\frac{(7 \cdot 2)}{(5 \cdot 2)} + \frac{(- 1 \cdot 5)}{(2 \cdot 5)}) k = (\frac{1}{2} \cdot k - 2) - \frac{1}{2} k$

Multiply the denominators:

$(\frac{(7 \cdot 2)}{10} + \frac{(- 1 \cdot 5)}{10}) k = (\frac{1}{2} \cdot k - 2) - \frac{1}{2} k$

Multiply the numerators:

$(\frac{14}{10} + \frac{- 5}{10}) k = (\frac{1}{2} \cdot k - 2) - \frac{1}{2} k$

Combine the fractions:

$\frac{(14 - 5)}{10} \cdot k = (\frac{1}{2} \cdot k - 2) - \frac{1}{2} k$

Combine the numerators:

$\frac{9}{10} \cdot k = (\frac{1}{2} \cdot k - 2) - \frac{1}{2} k$

Group like terms:

$\frac{9}{10} \cdot k = (\frac{1}{2} \cdot k + \frac{- 1}{2} k) - 2$

Combine the fractions:

$\frac{9}{10} \cdot k = \frac{(1 - 1)}{2} k - 2$

Combine the numerators:

$\frac{9}{10} \cdot k = \frac{0}{2} k - 2$

Reduce the zero numerator:

$\frac{9}{10} k = 0 k - 2$

Simplify the arithmetic:

$\frac{9}{10} k = - 2$

Multiply both sides by inverse fraction :

$(\frac{9}{10} k) \cdot \frac{10}{9} = - 2 \cdot \frac{10}{9}$

Group like terms:

$(\frac{9}{10} \cdot \frac{10}{9}) k = - 2 \cdot \frac{10}{9}$

Multiply the coefficients:

$\frac{(9 \cdot 10)}{(10 \cdot 9)} k = - 2 \cdot \frac{10}{9}$

Simplify the fraction:

$k = - 2 \cdot \frac{10}{9}$

Multiply the fraction(s):

$k = \frac{(- 2 \cdot 10)}{9}$

Simplify the arithmetic:

$k = \frac{- 20}{9}$

18 additional steps

$\frac{7}{5} k = - (\frac{1}{2} k - 2)$

Expand the parentheses:

$\frac{7}{5} \cdot k = \frac{- 1}{2} k + 2$

Add to both sides:

$(\frac{7}{5} k) + \frac{1}{2} \cdot k = (\frac{- 1}{2} k + 2) + \frac{1}{2} k$

Group the coefficients:

$(\frac{7}{5} + \frac{1}{2}) k = (\frac{- 1}{2} \cdot k + 2) + \frac{1}{2} k$

Find the lowest common denominator:

$(\frac{(7 \cdot 2)}{(5 \cdot 2)} + \frac{(1 \cdot 5)}{(2 \cdot 5)}) k = (\frac{- 1}{2} \cdot k + 2) + \frac{1}{2} k$

Multiply the denominators:

$(\frac{(7 \cdot 2)}{10} + \frac{(1 \cdot 5)}{10}) k = (\frac{- 1}{2} \cdot k + 2) + \frac{1}{2} k$

Multiply the numerators:

$(\frac{14}{10} + \frac{5}{10}) k = (\frac{- 1}{2} \cdot k + 2) + \frac{1}{2} k$

Combine the fractions:

$\frac{(14 + 5)}{10} \cdot k = (\frac{- 1}{2} \cdot k + 2) + \frac{1}{2} k$

Combine the numerators:

$\frac{19}{10} \cdot k = (\frac{- 1}{2} \cdot k + 2) + \frac{1}{2} k$

Group like terms:

$\frac{19}{10} \cdot k = (\frac{- 1}{2} \cdot k + \frac{1}{2} k) + 2$

Combine the fractions:

$\frac{19}{10} \cdot k = \frac{(- 1 + 1)}{2} k + 2$

Combine the numerators:

$\frac{19}{10} \cdot k = \frac{0}{2} k + 2$

Reduce the zero numerator:

$\frac{19}{10} k = 0 k + 2$

Simplify the arithmetic:

$\frac{19}{10} k = 2$

Multiply both sides by inverse fraction :

$(\frac{19}{10} k) \cdot \frac{10}{19} = 2 \cdot \frac{10}{19}$

Group like terms:

$(\frac{19}{10} \cdot \frac{10}{19}) k = 2 \cdot \frac{10}{19}$

Multiply the coefficients:

$\frac{(19 \cdot 10)}{(10 \cdot 19)} k = 2 \cdot \frac{10}{19}$

Simplify the fraction:

$k = 2 \cdot \frac{10}{19}$

Multiply the fraction(s):

$k = \frac{(2 \cdot 10)}{19}$

Simplify the arithmetic:

$k = \frac{20}{19}$

3. List the solutions

$k = - \frac{20}{9}, \frac{20}{19}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | \frac{7}{5} k |$
$y = | \frac{1}{2} k - 2 |$
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 k

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 k

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved