Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

k = 0, 0

k=0 , 0

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
$| 6 k | = | 7 k |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 6 k \| = \| 7 k \|$
$x = + y$	$(6 k) = (7 k)$
$x = - y$	$(6 k) = - (7 k)$
$+ x = y$	$(6 k) = (7 k)$
$- x = y$	$- (6 k) = (7 k)$

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:

$\| x \| = \| y \|$	$\| 6 k \| = \| 7 k \|$
$x = + y$ , $+ x = y$	$(6 k) = (7 k)$
$x = - y$ , $- x = y$	$(6 k) = - (7 k)$

2. Solve the two equations for k

5 additional steps

$6 k = 7 k$

Subtract from both sides:

$(6 k) - 7 k = (7 k) - 7 k$

Simplify the arithmetic:

$- k = (7 k) - 7 k$

Simplify the arithmetic:

$- k = 0$

Multiply both sides by :

$- k \cdot - 1 = 0 \cdot - 1$

Remove the one(s):

$k = 0 \cdot - 1$

Multiply by zero:

$k = 0$

11 additional steps

$6 k = - 7 k$

Divide both sides by :

$\frac{(6 k)}{6} = \frac{(- 7 k)}{6}$

Simplify the fraction:

$k = \frac{(- 7 k)}{6}$

Add to both sides:

$k + \frac{7}{6} \cdot k = (\frac{(- 7 k)}{6}) + \frac{7}{6} k$

Group the coefficients:

$(1 + \frac{7}{6}) k = (\frac{(- 7 k)}{6}) + \frac{7}{6} k$

Convert the integer into a fraction:

$(\frac{6}{6} + \frac{7}{6}) k = (\frac{(- 7 k)}{6}) + \frac{7}{6} k$

Combine the fractions:

$\frac{(6 + 7)}{6} \cdot k = (\frac{(- 7 k)}{6}) + \frac{7}{6} k$

Combine the numerators:

$\frac{13}{6} \cdot k = (\frac{(- 7 k)}{6}) + \frac{7}{6} k$

Combine the fractions:

$\frac{13}{6} \cdot k = \frac{(- 7 + 7)}{6} k$

Combine the numerators:

$\frac{13}{6} \cdot k = \frac{0}{6} k$

Reduce the zero numerator:

$\frac{13}{6} k = 0 k$

Simplify the arithmetic:

$\frac{13}{6} k = 0$

Divide both sides by the coefficient:

$k = 0$

3. List the solutions

$k = 0, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 6 k |$
$y = | 7 k |$
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