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Solution - Absolute value equations

Exact form:

x = \frac{8}{3}, - 6

x=\frac{8}{3} , -6

Mixed number form:

x = 2 \frac{2}{3}, - 6

x=2\frac{2}{3} , -6

Decimal form:

x = 2.667, - 6

x=2.667 , -6

See steps

Step-by-step explanation

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

$| 2 x - 1 | + | x - 7 | = 0$

Add $- | x - 7 |$ to both sides of the equation:

$| 2 x - 1 | + | x - 7 | - | x - 7 | = - | x - 7 |$

Simplify the arithmetic

$| 2 x - 1 | = - | x - 7 |$

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
$| 2 x - 1 | = - | x - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = - \| x - 7 \|$
$x = + y$	$(2 x - 1) = - (x - 7)$
$x = - y$	$(2 x - 1) = - - (x - 7)$
$+ x = y$	$(2 x - 1) = - (x - 7)$
$- x = y$	$- (2 x - 1) = - (x - 7)$

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:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = - \| x - 7 \|$
$x = + y$ , $+ x = y$	$(2 x - 1) = - (x - 7)$
$x = - y$ , $- x = y$	$(2 x - 1) = - - (x - 7)$

3. Solve the two equations for x

10 additional steps

$(2 x - 1) = - (x - 7)$

Expand the parentheses:

$(2 x - 1) = - x + 7$

Add to both sides:

$(2 x - 1) + x = (- x + 7) + x$

Group like terms:

$(2 x + x) - 1 = (- x + 7) + x$

Simplify the arithmetic:

$3 x - 1 = (- x + 7) + x$

Group like terms:

$3 x - 1 = (- x + x) + 7$

Simplify the arithmetic:

$3 x - 1 = 7$

Add to both sides:

$(3 x - 1) + 1 = 7 + 1$

Simplify the arithmetic:

$3 x = 7 + 1$

Simplify the arithmetic:

$3 x = 8$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{8}{3}$

Simplify the fraction:

$x = \frac{8}{3}$

8 additional steps

$(2 x - 1) = - (- (x - 7))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(2 x - 1) = x - 7$

Subtract from both sides:

$(2 x - 1) - x = (x - 7) - x$

Group like terms:

$(2 x - x) - 1 = (x - 7) - x$

Simplify the arithmetic:

$x - 1 = (x - 7) - x$

Group like terms:

$x - 1 = (x - x) - 7$

Simplify the arithmetic:

$x - 1 = - 7$

Add to both sides:

$(x - 1) + 1 = - 7 + 1$

Simplify the arithmetic:

$x = - 7 + 1$

Simplify the arithmetic:

$x = - 6$

4. List the solutions

$x = \frac{8}{3}, - 6$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 1 |$
$y = - | x - 7 |$
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