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High School Math : Solving Absolute Value Equations

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Example Questions

Example Question #1 : Absolute Value

$2x + \left | x - 3 \right | = -12$

Possible Answers:

$x= \{-15, -3\}$

x = -9

x = -3

x=-15

x= -6

Correct answer:

x=-15

Explanation:

Notice that the equation $2x + \left | x - 3 \right | = -12$ has an term both inside and outside the absolute value expression.

Since the absolute value expression will always produce a positive number and the right side of the equation is negative, a negative number must be added to the result of the absolute value expression to satisfy the equation. Therefore the term outside of the absolute value expression (in this case ) must be negative (meaning must be negative).

Since will be a negative number, the expression within the absolute value x-3 will also be negative (before the absolute value is taken). It is thus possible to convert the original equation into an equation that treats the absolute value as a parenthetical expression that will be multiplied by , since any negative value becomes its opposite when taking the absolute value.

$2x + \left | x - 3 \right | = -12$

$\rightarrow 2x+ (x-3)(-1) = -12$

Simplifying and solving this equation for gives the answer:

$\rightarrow 2x -x+3 = -12$

$\rightarrow x + 3 = -12$

$\rightarrow x = -15$

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Example Question #2 : Absolute Value

|x|=12

What are the possible values for ?

Possible Answers:

$-\sqrt{12},\sqrt{12}$

-12, 12

144

Correct answer:

-12, 12

Explanation:

The absolute value measures the distance from zero to the given point.

In this case, since |x|=12 , x=12 or x=-12 , as both values are twelve units away from zero.

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Example Question #3 : Absolute Value

$\textup{What are all possible values of }b\textup{ when }\left | 3b-4\right |=11\textup{ ?}$

Possible Answers:

$\frac{7}{3}, -5$

$5, -\frac{3}{7}$

$\frac{19}{3}, -5$

$5, -\frac{7}{3}$

$5, -\frac{11}{3}$

Correct answer:

$5, -\frac{7}{3}$

Explanation:

$\textup{Set up two equations and solve for both solutions:}$

$3b-4=11\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;3b-4 = -11$

$3b = 15\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \; \; \; \; \; \; 3b=-7$

$b = 5\;\;\:\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;b=-\frac{7}{3}$

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Example Question #1 : Mathematical Relationships And Basic Graphs

Solve:

$|x-5|\leqslant 0$

Possible Answers:

All real numbers

No solution

$x=5 \; or \; x=-5$

$-5\leqslant x\leqslant 5$

x=5

Correct answer:

x=5

Explanation:

The absolute value can never be negative, so the equation is ONLY valid at zero.

The equation to solve becomes x-5=0 .

x=5

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Example Question #1 : Solving Absolute Value Equations

Solve for :

$\left | 2x-5\right |=3x$

Possible Answers:

x=-5,0

x=-1,5

x=1

x=-5

x=5

Correct answer:

x=1

Explanation:

To solve absolute value equations, we must understand that the absoute value function makes a value positive. So when we are solving these problems, we must consider two scenarios, one where the value is positive and one where the value is negative.

$\left | 2x-5\right |=3x$

2x-5=3x

and

2x-5=-(3x)

This gives us:

-5=x and 5x=5

$x=-5\ and\ 1$

However, this question has an outside of the absolute value expression, in this case . Thus, any negative value of will make the right side of the equation equal to a negative number, which cannot be true for an absolute value expression. Thus, x=-5 is an extraneous solution, as $\left | 2x-5\right |$ cannot equal a negative number.

Our final solution is then

x=1

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Example Question #1 : Absolute Value

Solve for .

$3\left | x+2 \right |=18$

Possible Answers:

x=8,4

x=-8,-4

x=4

x=-8,4

x=8,-4

Correct answer:

x=-8,4

Explanation:

$3\left | x+2 \right |=18$

Divide both sides by 3.

$\left | x+2 \right |=6$

Consider both the negative and positive values for the absolute value term.

$x+2=6\ or\ x+2=-6$

Subtract 2 from both sides to solve both scenarios for .

$x=-8\ or\ x=4$

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High School Math : Solving Absolute Value Equations

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Absolute Value

Example Question #2 : Absolute Value

Example Question #3 : Absolute Value

Example Question #1 : Mathematical Relationships And Basic Graphs

Example Question #1 : Solving Absolute Value Equations

Example Question #1 : Absolute Value

All High School Math Resources

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