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Solution - Absolute value equations

Exact form:

f = - 1

f=-1

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
$| f - 2 | = | f + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| f - 2 \| = \| f + 4 \|$
$x = + y$	$(f - 2) = (f + 4)$
$x = - y$	$(f - 2) = - (f + 4)$
$+ x = y$	$(f - 2) = (f + 4)$
$- x = y$	$- (f - 2) = (f + 4)$

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:

$\| x \| = \| y \|$	$\| f - 2 \| = \| f + 4 \|$
$x = + y$ , $+ x = y$	$(f - 2) = (f + 4)$
$x = - y$ , $- x = y$	$(f - 2) = - (f + 4)$

2. Solve the two equations for f

5 additional steps

$(f - 2) = (f + 4)$

Subtract from both sides:

$(f - 2) - f = (f + 4) - f$

Group like terms:

$(f - f) - 2 = (f + 4) - f$

Simplify the arithmetic:

$- 2 = (f + 4) - f$

Group like terms:

$- 2 = (f - f) + 4$

Simplify the arithmetic:

$- 2 = 4$

The statement is false:

$- 2 = 4$

The equation is false so it has no solution.

11 additional steps

$(f - 2) = - (f + 4)$

Expand the parentheses:

$(f - 2) = - f - 4$

Add to both sides:

$(f - 2) + f = (- f - 4) + f$

Group like terms:

$(f + f) - 2 = (- f - 4) + f$

Simplify the arithmetic:

$2 f - 2 = (- f - 4) + f$

Group like terms:

$2 f - 2 = (- f + f) - 4$

Simplify the arithmetic:

$2 f - 2 = - 4$

Add to both sides:

$(2 f - 2) + 2 = - 4 + 2$

Simplify the arithmetic:

$2 f = - 4 + 2$

Simplify the arithmetic:

$2 f = - 2$

Divide both sides by :

$\frac{(2 f)}{2} = \frac{- 2}{2}$

Simplify the fraction:

$f = \frac{- 2}{2}$

Simplify the fraction:

$f = - 1$

3. Graph

Each line represents the function of one side of the equation:
$y = | f - 2 |$
$y = | f + 4 |$
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 f

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 f

3. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved