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Solution - Absolute value equations

Exact form:

p = \frac{3}{2}, \frac{1}{2}

p=\frac{3}{2} , \frac{1}{2}

Mixed number form:

p = 1 \frac{1}{2}, \frac{1}{2}

p=1\frac{1}{2} , \frac{1}{2}

Decimal form:

p = 1.5, 0.5

p=1.5 , 0.5

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

$\| x \| = \| y \|$	$\| 4 p - 3 \| = \| 2 p \|$
$x = + y$	$(4 p - 3) = (2 p)$
$x = - y$	$(4 p - 3) = - (2 p)$
$+ x = y$	$(4 p - 3) = (2 p)$
$- x = y$	$- (4 p - 3) = (2 p)$

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:

$\| x \| = \| y \|$	$\| 4 p - 3 \| = \| 2 p \|$
$x = + y$ , $+ x = y$	$(4 p - 3) = (2 p)$
$x = - y$ , $- x = y$	$(4 p - 3) = - (2 p)$

2. Solve the two equations for p

8 additional steps

$(4 p - 3) = 2 p$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$2 p - 3 = 0$

Add to both sides:

$(2 p - 3) + 3 = 0 + 3$

Simplify the arithmetic:

$2 p = 0 + 3$

Simplify the arithmetic:

$2 p = 3$

Divide both sides by :

$\frac{(2 p)}{2} = \frac{3}{2}$

Simplify the fraction:

$p = \frac{3}{2}$

9 additional steps

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

Add to both sides:

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

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

$6 p = ((- 2 p) + 3) + 2 p$

Group like terms:

$6 p = (- 2 p + 2 p) + 3$

Simplify the arithmetic:

$6 p = 3$

Divide both sides by :

$\frac{(6 p)}{6} = \frac{3}{6}$

Simplify the fraction:

$p = \frac{3}{6}$

Find the greatest common factor of the numerator and denominator:

$p = \frac{(1 \cdot 3)}{(2 \cdot 3)}$

Factor out and cancel the greatest common factor:

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

3. List the solutions

$p = \frac{3}{2}, \frac{1}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 p - 3 |$
$y = | 2 p |$
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 p

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 p

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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