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Solution - Absolute value equations

Exact form:

x = 3, 3

x=3 , 3

See steps

Step-by-step explanation

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

$| 5 x - 15 | + | 6 x - 18 | = 0$

Add $- | 6 x - 18 |$ to both sides of the equation:

$| 5 x - 15 | + | 6 x - 18 | - | 6 x - 18 | = - | 6 x - 18 |$

Simplify the arithmetic

$| 5 x - 15 | = - | 6 x - 18 |$

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
$| 5 x - 15 | = - | 6 x - 18 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x - 15 \| = - \| 6 x - 18 \|$
$x = + y$	$(5 x - 15) = - (6 x - 18)$
$x = - y$	$(5 x - 15) = - - (6 x - 18)$
$+ x = y$	$(5 x - 15) = - (6 x - 18)$
$- x = y$	$- (5 x - 15) = - (6 x - 18)$

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:

$\| x \| = \| y \|$	$\| 5 x - 15 \| = - \| 6 x - 18 \|$
$x = + y$ , $+ x = y$	$(5 x - 15) = - (6 x - 18)$
$x = - y$ , $- x = y$	$(5 x - 15) = - - (6 x - 18)$

3. Solve the two equations for x

12 additional steps

$(5 x - 15) = - (6 x - 18)$

Expand the parentheses:

$(5 x - 15) = - 6 x + 18$

Add to both sides:

$(5 x - 15) + 6 x = (- 6 x + 18) + 6 x$

Group like terms:

$(5 x + 6 x) - 15 = (- 6 x + 18) + 6 x$

Simplify the arithmetic:

$11 x - 15 = (- 6 x + 18) + 6 x$

Group like terms:

$11 x - 15 = (- 6 x + 6 x) + 18$

Simplify the arithmetic:

$11 x - 15 = 18$

Add to both sides:

$(11 x - 15) + 15 = 18 + 15$

Simplify the arithmetic:

$11 x = 18 + 15$

Simplify the arithmetic:

$11 x = 33$

Divide both sides by :

$\frac{(11 x)}{11} = \frac{33}{11}$

Simplify the fraction:

$x = \frac{33}{11}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 11)}{(1 \cdot 11)}$

Factor out and cancel the greatest common factor:

$x = 3$

11 additional steps

$(5 x - 15) = - (- (6 x - 18))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(5 x - 15) = 6 x - 18$

Subtract from both sides:

$(5 x - 15) - 6 x = (6 x - 18) - 6 x$

Group like terms:

$(5 x - 6 x) - 15 = (6 x - 18) - 6 x$

Simplify the arithmetic:

$- x - 15 = (6 x - 18) - 6 x$

Group like terms:

$- x - 15 = (6 x - 6 x) - 18$

Simplify the arithmetic:

$- x - 15 = - 18$

Add to both sides:

$(- x - 15) + 15 = - 18 + 15$

Simplify the arithmetic:

$- x = - 18 + 15$

Simplify the arithmetic:

$- x = - 3$

Multiply both sides by :

$- x \cdot - 1 = - 3 \cdot - 1$

Remove the one(s):

$x = - 3 \cdot - 1$

Simplify the arithmetic:

$x = 3$

4. List the solutions

$x = 3, 3$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 5 x - 15 |$
$y = - | 6 x - 18 |$
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 x

4. List the solutions

5. 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 x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved