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Solution - Absolute value equations

Exact form:

l = 2

l=2

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
$| - l + 1 | = | - l + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - l + 1 \| = \| - l + 3 \|$
$x = + y$	$(- l + 1) = (- l + 3)$
$x = - y$	$(- l + 1) = - (- l + 3)$
$+ x = y$	$(- l + 1) = (- l + 3)$
$- x = y$	$- (- l + 1) = (- l + 3)$

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:

$\| x \| = \| y \|$	$\| - l + 1 \| = \| - l + 3 \|$
$x = + y$ , $+ x = y$	$(- l + 1) = (- l + 3)$
$x = - y$ , $- x = y$	$(- l + 1) = - (- l + 3)$

2. Solve the two equations for l

5 additional steps

$(- l + 1) = (- l + 3)$

Add to both sides:

$(- l + 1) + l = (- l + 3) + l$

Group like terms:

$(- l + l) + 1 = (- l + 3) + l$

Simplify the arithmetic:

$1 = (- l + 3) + l$

Group like terms:

$1 = (- l + l) + 3$

Simplify the arithmetic:

$1 = 3$

The statement is false:

$1 = 3$

The equation is false so it has no solution.

14 additional steps

$(- l + 1) = - (- l + 3)$

Expand the parentheses:

$(- l + 1) = l - 3$

Subtract from both sides:

$(- l + 1) - l = (l - 3) - l$

Group like terms:

$(- l - l) + 1 = (l - 3) - l$

Simplify the arithmetic:

$- 2 l + 1 = (l - 3) - l$

Group like terms:

$- 2 l + 1 = (l - l) - 3$

Simplify the arithmetic:

$- 2 l + 1 = - 3$

Subtract from both sides:

$(- 2 l + 1) - 1 = - 3 - 1$

Simplify the arithmetic:

$- 2 l = - 3 - 1$

Simplify the arithmetic:

$- 2 l = - 4$

Divide both sides by :

$\frac{(- 2 l)}{- 2} = \frac{- 4}{- 2}$

Cancel out the negatives:

$\frac{2 l}{2} = \frac{- 4}{- 2}$

Simplify the fraction:

$l = \frac{- 4}{- 2}$

Cancel out the negatives:

$l = \frac{4}{2}$

Find the greatest common factor of the numerator and denominator:

$l = \frac{(2 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$l = 2$

3. Graph

Each line represents the function of one side of the equation:
$y = | - l + 1 |$
$y = | - l + 3 |$
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 l

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 l

3. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved