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Algebra II : Mathematical Relationships and Basic Graphs

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Example Questions

Example Question #21 : Absolute Value

Solve:

$\left | x+7\right | = 3$

Possible Answers:

x = -4

x = -3

x = 3

x = -10

x = -4 or x = -10

Correct answer:

x = -4 or x = -10

Explanation:

To solve:

$\left | x+7\right | = 3$

1. To clear the absolute value sign, we must split the equation to two possible solution. One solution for the possibility of a positive sign and another for the possibility of a negative sign:

x+7 = 3 or x+7 = -3

2. solve each equation for :

x = -4 or x = -10

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Example Question #26 : Absolute Value

Solve the following equation for b:

$\left | 5b-13\right |+6=8$

Possible Answers:

$\frac{11}{5}$

$b=3, \frac{11}{5}$

Correct answer:

$b=3, \frac{11}{5}$

Explanation:

Solve the following equation for b:

$\left | 5b-13\right |+6=8$

Let's begin by subtracting 6 from both sides:

$\left | 5b-13\right |=8-6$

$\left | 5b-13\right |=2$

Next, we can get rid of the absolute value signs and make our two equations:

5b-13=2

Next, add 13 to both sides

5b=2+13

5b=15

And divide by 5:

$b=\frac{15}{5}=3$

b=3

5b-13=-2

Next, add 13 to both sides

5b=11

And divide by 5:

$b=\frac{11}{5}$

So our answers are

$b=3, \frac{11}{5}$

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Example Question #21 : Solving Absolute Value Equations

Solve for x:

|2x-3|=-1

Possible Answers:

no solution

1,2

Correct answer:

no solution

Explanation:

Since the absolute value function only produces positive answers, an absolute value can never be equal to a negative number.

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Example Question #2152 : Mathematical Relationships And Basic Graphs

Solve the following equation:

$\small \left | x+4\right |=5$

Possible Answers:

$\small x=-1, x=9$

$\small \small x=-9, x=-1$

$\small x=-9, x=1$

$\small x=-9$

$\small x=1$

Correct answer:

$\small x=-9, x=1$

Explanation:

To solve this you need to set up two different equations then solve for x.

The first one is:

$\small x+4=5$ where $\small x=1$

The other equations is:

$\small x+4=-5$ where $\small x=-9$

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Example Question #29 : Absolute Value

Find values of which satisfy,

$2x - \left |3x+7 \right | = 4$

Possible Answers:

No solution

$x = 11, \frac{3}{5}$

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

$x = \frac{3}{5}, -\frac{3}{5}$

$x =- \frac{11}{5}, 11$

Correct answer:

No solution

Explanation:

$2x - \left |3x+7 \right | = 4$

Recall the general definition of absolute value,

$\left |A\right| = \begin{cases} &+A \text{ if } A\geq 0 \\ & -A\text{ if } A<0 \end{cases}$

We will attempt two cases for the absolute value,

|3x+7| = 3x+7

|3x+7| = -(3x+7)

Case 1:

Start by isolating the abolute value term,

$\left| 3x+7 \right| = 2x -4$

Replace |3x+7| with 3x+7 and solve for :

3x+7 = 2x-4

Case 2:

$\left| 3x+7 \right| = 2x -4$

Replace |3x+7| with -(3x+7) and solve for :

-(3x+7)=2x-4

-3x - 7 = 2x -4

-5x =3

$x = -\frac{3}{5}$

Testing the Solutions

Whenever solving equations with an absolute value it is crucuial to check if the solutions work in the original equation. It often occurs that one or both of the solutions will not satisfy the original equation.

For instance, if we test x = -11 in the original equation we get,

$\left| 3(-11)+7 \right| = 2(-11) -4$

26 = -26

This is clearly not true, since both sides are not equal, this rules out x=-11 as a solution. Similiarly, using $x= -\frac{3}{5}$ you can show that it also fails. Therefore, there is no solution.

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Example Question #23 : Absolute Value

What are all the possible values of that fulfill the equation below?

$\left | 3x-7\right |=15$

Possible Answers:

$\frac{8}{3}$ and $\frac{22}{3}$

$\frac{22}{3}$ only

$\frac{22}{3}$ and $-\frac{8}{3}$

$-\frac{22}{3}$ and $\frac{8}{3}$

$-\frac{8}{3}$ and $\frac{8}{3}$

Correct answer:

$\frac{22}{3}$ and $-\frac{8}{3}$

Explanation:

If $\left | 3x-7\right |=15$ , then 3x-7=15 or 3x-7=-15

Solve each of those equations to find the possible values for x.

$x=\frac{22}{3}$ or $x=-\frac{8}{3}$

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Example Question #31 : Absolute Value

Which of the following is in the set of all possible values for y in the equation $3\left | 5-y\right |+8=29$ ?

Possible Answers:

$\left \{ 12\right \}$

The empty set.

$\left \{-2, 12 \right \}$

$\left \{ 2\right \}$

$\left \{ -12,2\right \}$

Correct answer:

$\left \{-2, 12 \right \}$

Explanation:

First isolate the absolute value:

$3\left | 5-y\right |+8=29$

$3\left | 5-y\right |=21$

$\left | 5-y\right |=7$

There are two possible values for 5-y : 7 and -7.

5-y=7 or 5-y=-7

y=-2 or y=12

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Example Question #2151 : Mathematical Relationships And Basic Graphs

What is the set of all values for x that satisfy the equation $-2\left | 2x+5\right |=22$ ?

Possible Answers:

$\left \{ -8,3\right \}$

The empty set.

$\left \{ -\frac{27}{4},\frac{17}{4}\right \}$

$\left \{ -\frac{17}{4}, \frac{27}{4}\right \}$

$\left \{ -3, 8\right \}$

Correct answer:

The empty set.

Explanation:

First isolate the absolute value.

$-2\left | 2x+5\right |=22$

$\left | 2x+5\right |=-11$

There are no real numbers for which you can take the absolute value and get a negative number. Therefore, there are no values for x that will satisfy the equation.

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Example Question #33 : Absolute Value

Which of the following is the set of all values for that satisfy the equation $5\left |2-3x \right |-33=-8$ ?

Possible Answers:

$\left \{ -5,5\right \}$

The empty set.

$\left \{ -1,\frac{7}{3}\right \}$

$\left \{ -\frac{7}{3}, 1\right \}$

$\left \{ -\frac{31}{3}\right \}$

Correct answer:

$\left \{ -1,\frac{7}{3}\right \}$

Explanation:

First isolate the absolute value.

$5\left |2-3x \right |-33=-8$

$5\left |2-3x \right |=25$

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

The two possible values for 2-3x are 5 and -5.

2-3x=5 or 2-3x=-5

x=-1 or $x=\frac{7}{3}$

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Example Question #31 : Absolute Value

Solve for .

$2\left | x\right |+14=16$

Possible Answers:

$\pm1$

Correct answer:

$\pm1$

Explanation:

When dealing with absolute value equations, we need to deal with negative values as well.

$2\left | x\right |+14=16$ Subtract on both sides.

$2\left | x\right |=2$ Divide on both sides.

$\left | x\right |=1$

$x=\pm1$

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #21 : Absolute Value

Example Question #26 : Absolute Value

Example Question #21 : Solving Absolute Value Equations

Example Question #2152 : Mathematical Relationships And Basic Graphs

Example Question #29 : Absolute Value

Case 1:

Case 2:

Testing the Solutions

Example Question #23 : Absolute Value

Example Question #31 : Absolute Value

Example Question #2151 : Mathematical Relationships And Basic Graphs

Example Question #33 : Absolute Value

Example Question #31 : Absolute Value

All Algebra II Resources

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