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Solution - Absolute value equations

Exact form:

k = 3, \frac{4}{3}

k=3 , \frac{4}{3}

Mixed number form:

k = 3, 1 \frac{1}{3}

k=3 , 1\frac{1}{3}

Decimal form:

k = 3, 1.333

k=3 , 1.333

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
$5 | k - 2 | = | k + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$5 \| k - 2 \| = \| k + 2 \|$
$x = + y$	$5 (k - 2) = (k + 2)$
$x = - y$	$5 (k - 2) = - (k + 2)$
$+ x = y$	$5 (k - 2) = (k + 2)$
$- x = y$	$5 (- (k - 2)) = (k + 2)$

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:

$\| x \| = \| y \|$	$5 \| k - 2 \| = \| k + 2 \|$
$x = + y$ , $+ x = y$	$5 (k - 2) = (k + 2)$
$x = - y$ , $- x = y$	$5 (k - 2) = - (k + 2)$

2. Solve the two equations for k

13 additional steps

$5 \cdot (k - 2) = (k + 2)$

Expand the parentheses:

$5 k + 5 \cdot - 2 = (k + 2)$

Simplify the arithmetic:

$5 k - 10 = (k + 2)$

Subtract from both sides:

$(5 k - 10) - k = (k + 2) - k$

Group like terms:

$(5 k - k) - 10 = (k + 2) - k$

Simplify the arithmetic:

$4 k - 10 = (k + 2) - k$

Group like terms:

$4 k - 10 = (k - k) + 2$

Simplify the arithmetic:

$4 k - 10 = 2$

Add to both sides:

$(4 k - 10) + 10 = 2 + 10$

Simplify the arithmetic:

$4 k = 2 + 10$

Simplify the arithmetic:

$4 k = 12$

Divide both sides by :

$\frac{(4 k)}{4} = \frac{12}{4}$

Simplify the fraction:

$k = \frac{12}{4}$

Find the greatest common factor of the numerator and denominator:

$k = \frac{(3 \cdot 4)}{(1 \cdot 4)}$

Factor out and cancel the greatest common factor:

$k = 3$

14 additional steps

$5 \cdot (k - 2) = - (k + 2)$

Expand the parentheses:

$5 k + 5 \cdot - 2 = - (k + 2)$

Simplify the arithmetic:

$5 k - 10 = - (k + 2)$

Expand the parentheses:

$5 k - 10 = - k - 2$

Add to both sides:

$(5 k - 10) + k = (- k - 2) + k$

Group like terms:

$(5 k + k) - 10 = (- k - 2) + k$

Simplify the arithmetic:

$6 k - 10 = (- k - 2) + k$

Group like terms:

$6 k - 10 = (- k + k) - 2$

Simplify the arithmetic:

$6 k - 10 = - 2$

Add to both sides:

$(6 k - 10) + 10 = - 2 + 10$

Simplify the arithmetic:

$6 k = - 2 + 10$

Simplify the arithmetic:

$6 k = 8$

Divide both sides by :

$\frac{(6 k)}{6} = \frac{8}{6}$

Simplify the fraction:

$k = \frac{8}{6}$

Find the greatest common factor of the numerator and denominator:

$k = \frac{(4 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$k = \frac{4}{3}$

3. List the solutions

$k = 3, \frac{4}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 5 | k - 2 |$
$y = | k + 2 |$
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 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

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