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Solution - Absolute value equations

Exact form:

d = \frac{3}{4}, - 3

d=\frac{3}{4} , -3

Decimal form:

d = 0.75, - 3

d=0.75 , -3

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
$| 4 d - 3 | = | - 4 d + 3 |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for d

11 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$8 d - 3 = (- 4 d + 3) + 4 d$

Group like terms:

$8 d - 3 = (- 4 d + 4 d) + 3$

Simplify the arithmetic:

$8 d - 3 = 3$

Add to both sides:

$(8 d - 3) + 3 = 3 + 3$

Simplify the arithmetic:

$8 d = 3 + 3$

Simplify the arithmetic:

$8 d = 6$

Divide both sides by :

$\frac{(8 d)}{8} = \frac{6}{8}$

Simplify the fraction:

$d = \frac{6}{8}$

Find the greatest common factor of the numerator and denominator:

$d = \frac{(3 \cdot 2)}{(4 \cdot 2)}$

Factor out and cancel the greatest common factor:

$d = \frac{3}{4}$

5 additional steps

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

Expand the parentheses:

$(4 d - 3) = 4 d - 3$

Subtract from both sides:

$(4 d - 3) - 4 d = (4 d - 3) - 4 d$

Group like terms:

$(4 d - 4 d) - 3 = (4 d - 3) - 4 d$

Simplify the arithmetic:

$- 3 = (4 d - 3) - 4 d$

Group like terms:

$- 3 = (4 d - 4 d) - 3$

Simplify the arithmetic:

$- 3 = - 3$

3. List the solutions

$d = \frac{3}{4}, - 3$
(2 solution(s))

4. Graph

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

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 d

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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