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Solution - Absolute value equations

Exact form:

p = - \frac{1}{2}

p=-\frac{1}{2}

Decimal form:

p = - 0.5

p=-0.5

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| - p + 3 | - | - p - 4 | = 0$

Add $| - p - 4 |$ to both sides of the equation:

$| - p + 3 | - | - p - 4 | + | - p - 4 | = | - p - 4 |$

Simplify the arithmetic

$| - p + 3 | = | - p - 4 |$

2. 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
$| - p + 3 | = | - p - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - p + 3 \| = \| - p - 4 \|$
$x = + y$	$(- p + 3) = (- p - 4)$
$x = - y$	$(- p + 3) = (- (- p - 4))$
$+ x = y$	$(- p + 3) = (- p - 4)$
$- x = y$	$- (- p + 3) = (- p - 4)$

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:

$\| x \| = \| y \|$	$\| - p + 3 \| = \| - p - 4 \|$
$x = + y$ , $+ x = y$	$(- p + 3) = (- p - 4)$
$x = - y$ , $- x = y$	$(- p + 3) = (- (- p - 4))$

3. Solve the two equations for p

5 additional steps

$(- p + 3) = (- p - 4)$

Add to both sides:

$(- p + 3) + p = (- p - 4) + p$

Group like terms:

$(- p + p) + 3 = (- p - 4) + p$

Simplify the arithmetic:

$3 = (- p - 4) + p$

Group like terms:

$3 = (- p + p) - 4$

Simplify the arithmetic:

$3 = - 4$

The statement is false:

$3 = - 4$

The equation is false so it has no solution.

12 additional steps

$(- p + 3) = - (- p - 4)$

Expand the parentheses:

$(- p + 3) = p + 4$

Subtract from both sides:

$(- p + 3) - p = (p + 4) - p$

Group like terms:

$(- p - p) + 3 = (p + 4) - p$

Simplify the arithmetic:

$- 2 p + 3 = (p + 4) - p$

Group like terms:

$- 2 p + 3 = (p - p) + 4$

Simplify the arithmetic:

$- 2 p + 3 = 4$

Subtract from both sides:

$(- 2 p + 3) - 3 = 4 - 3$

Simplify the arithmetic:

$- 2 p = 4 - 3$

Simplify the arithmetic:

$- 2 p = 1$

Divide both sides by :

$\frac{(- 2 p)}{- 2} = \frac{1}{- 2}$

Cancel out the negatives:

$\frac{2 p}{2} = \frac{1}{- 2}$

Simplify the fraction:

$p = \frac{1}{- 2}$

Move the negative sign from the denominator to the numerator:

$p = \frac{- 1}{2}$

4. Graph

Each line represents the function of one side of the equation:
$y = | - p + 3 |$
$y = | - p - 4 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for p

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for p

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved