Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

d = 1, 9

d=1 , 9

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
$| d + 3 | = | - 2 d + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| d + 3 \| = \| - 2 d + 6 \|$
$x = + y$	$(d + 3) = (- 2 d + 6)$
$x = - y$	$(d + 3) = - (- 2 d + 6)$
$+ x = y$	$(d + 3) = (- 2 d + 6)$
$- x = y$	$- (d + 3) = (- 2 d + 6)$

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:

$\| x \| = \| y \|$	$\| d + 3 \| = \| - 2 d + 6 \|$
$x = + y$ , $+ x = y$	$(d + 3) = (- 2 d + 6)$
$x = - y$ , $- x = y$	$(d + 3) = - (- 2 d + 6)$

2. Solve the two equations for d

10 additional steps

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$3 d + 3 = 6$

Subtract from both sides:

$(3 d + 3) - 3 = 6 - 3$

Simplify the arithmetic:

$3 d = 6 - 3$

Simplify the arithmetic:

$3 d = 3$

Divide both sides by :

$\frac{(3 d)}{3} = \frac{3}{3}$

Simplify the fraction:

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

Simplify the fraction:

$d = 1$

11 additional steps

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

Expand the parentheses:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- d + 3 = - 6$

Subtract from both sides:

$(- d + 3) - 3 = - 6 - 3$

Simplify the arithmetic:

$- d = - 6 - 3$

Simplify the arithmetic:

$- d = - 9$

Multiply both sides by :

$- d \cdot - 1 = - 9 \cdot - 1$

Remove the one(s):

$d = - 9 \cdot - 1$

Simplify the arithmetic:

$d = 9$

3. List the solutions

$d = 1, 9$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | d + 3 |$
$y = | - 2 d + 6 |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

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

Latest Related Drills Solved