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Solution - Absolute value equations

Exact form:

k = 6

k=6

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
$| - k + 7 | = | - k + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - k + 7 \| = \| - k + 5 \|$
$x = + y$	$(- k + 7) = (- k + 5)$
$x = - y$	$(- k + 7) = - (- k + 5)$
$+ x = y$	$(- k + 7) = (- k + 5)$
$- x = y$	$- (- k + 7) = (- k + 5)$

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:

$\| x \| = \| y \|$	$\| - k + 7 \| = \| - k + 5 \|$
$x = + y$ , $+ x = y$	$(- k + 7) = (- k + 5)$
$x = - y$ , $- x = y$	$(- k + 7) = - (- k + 5)$

2. Solve the two equations for k

5 additional steps

$(- k + 7) = (- k + 5)$

Add to both sides:

$(- k + 7) + k = (- k + 5) + k$

Group like terms:

$(- k + k) + 7 = (- k + 5) + k$

Simplify the arithmetic:

$7 = (- k + 5) + k$

Group like terms:

$7 = (- k + k) + 5$

Simplify the arithmetic:

$7 = 5$

The statement is false:

$7 = 5$

The equation is false so it has no solution.

14 additional steps

$(- k + 7) = - (- k + 5)$

Expand the parentheses:

$(- k + 7) = k - 5$

Subtract from both sides:

$(- k + 7) - k = (k - 5) - k$

Group like terms:

$(- k - k) + 7 = (k - 5) - k$

Simplify the arithmetic:

$- 2 k + 7 = (k - 5) - k$

Group like terms:

$- 2 k + 7 = (k - k) - 5$

Simplify the arithmetic:

$- 2 k + 7 = - 5$

Subtract from both sides:

$(- 2 k + 7) - 7 = - 5 - 7$

Simplify the arithmetic:

$- 2 k = - 5 - 7$

Simplify the arithmetic:

$- 2 k = - 12$

Divide both sides by :

$\frac{(- 2 k)}{- 2} = \frac{- 12}{- 2}$

Cancel out the negatives:

$\frac{2 k}{2} = \frac{- 12}{- 2}$

Simplify the fraction:

$k = \frac{- 12}{- 2}$

Cancel out the negatives:

$k = \frac{12}{2}$

Find the greatest common factor of the numerator and denominator:

$k = \frac{(6 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$k = 6$

3. Graph

Each line represents the function of one side of the equation:
$y = | - k + 7 |$
$y = | - k + 5 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved