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Solution - Absolute value equations

Exact form:

d = - 1.265, - 0.99

d=-1.265 , -0.99

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
$| 2.5 d + 1.8 | = | 7.4 d + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2.5 d + 1.8 \| = \| 7.4 d + 8 \|$
$x = + y$	$(2.5 d + 1.8) = (7.4 d + 8)$
$x = - y$	$(2.5 d + 1.8) = - (7.4 d + 8)$
$+ x = y$	$(2.5 d + 1.8) = (7.4 d + 8)$
$- x = y$	$- (2.5 d + 1.8) = (7.4 d + 8)$

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:

$\| x \| = \| y \|$	$\| 2.5 d + 1.8 \| = \| 7.4 d + 8 \|$
$x = + y$ , $+ x = y$	$(2.5 d + 1.8) = (7.4 d + 8)$
$x = - y$ , $- x = y$	$(2.5 d + 1.8) = - (7.4 d + 8)$

2. Solve the two equations for d

12 additional steps

$(2.5 d + 1.8) = (7.4 d + 8)$

Subtract from both sides:

$(2.5 d + 1.8) - 7.4 d = (7.4 d + 8) - 7.4 d$

Group like terms:

$(2.5 d - 7.4 d) + 1.8 = (7.4 d + 8) - 7.4 d$

Simplify the arithmetic:

$- 4.9 d + 1.8 = (7.4 d + 8) - 7.4 d$

Group like terms:

$- 4.9 d + 1.8 = (7.4 d - 7.4 d) + 8$

Simplify the arithmetic:

$- 4.9 d + 1.8 = 8$

Subtract from both sides:

$(- 4.9 d + 1.8) - 1.8 = 8 - 1.8$

Simplify the arithmetic:

$- 4.9 d = 8 - 1.8$

Simplify the arithmetic:

$- 4.9 d = 6.2$

Divide both sides by :

$\frac{(- 4.9 d)}{- 4.9} = \frac{6.2}{- 4.9}$

Cancel out the negatives:

$\frac{4.9 d}{4.9} = \frac{6.2}{- 4.9}$

Simplify the arithmetic:

$d = \frac{6.2}{- 4.9}$

Move the negative sign from the denominator to the numerator:

$d = \frac{- 6.2}{4.9}$

Simplify the arithmetic:

$d = - 1.2653$

11 additional steps

$(2.5 d + 1.8) = - (7.4 d + 8)$

Expand the parentheses:

$(2.5 d + 1.8) = - 7.4 d - 8$

Add to both sides:

$(2.5 d + 1.8) + 7.4 d = (- 7.4 d - 8) + 7.4 d$

Group like terms:

$(2.5 d + 7.4 d) + 1.8 = (- 7.4 d - 8) + 7.4 d$

Simplify the arithmetic:

$9.9 d + 1.8 = (- 7.4 d - 8) + 7.4 d$

Group like terms:

$9.9 d + 1.8 = (- 7.4 d + 7.4 d) - 8$

Simplify the arithmetic:

$9.9 d + 1.8 = - 8$

Subtract from both sides:

$(9.9 d + 1.8) - 1.8 = - 8 - 1.8$

Simplify the arithmetic:

$9.9 d = - 8 - 1.8$

Simplify the arithmetic:

$9.9 d = - 9.8$

Divide both sides by :

$\frac{(9.9 d)}{9.9} = \frac{- 9.8}{9.9}$

Simplify the arithmetic:

$d = \frac{- 9.8}{9.9}$

Simplify the arithmetic:

$d = - 0.9899$

3. List the solutions

$d = - 1.265, - 0.99$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2.5 d + 1.8 |$
$y = | 7.4 d + 8 |$
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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