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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #12 : Solving Absolute Value Equations

Solve for .

$\frac{x}{5}=\left | 3\right |$

Possible Answers:

-5, 5

15, -15

Correct answer:

15, -15

Explanation:

When solving for absolute values, remember there's an equation for positive value and another equation for negative value.

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

Multiply both sides by for both equations.

x=15, x=-15

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Example Question #13 : Solving Absolute Value Equations

Solve for .

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

Possible Answers:

$-\frac{3}{7}$

$\pm \frac{7}{3}$

$\frac{3}{7}$

$\pm \frac{3}{7}$

$\frac{7}{3}$

Correct answer:

$\pm \frac{3}{7}$

Explanation:

When solving for absolute values, remember there's an equation for positive value and another equation for negative value.

7x=3, 7x=-3

Divide both sides by for both equations.

$x=\pm \frac{3}{7}$

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Example Question #14 : Solving Absolute Value Equations

Solve:

$\left | 2x-1 \right |=13$

Possible Answers:

x=7, 6

x=-6

x=7,-6

x=7

x=-7,-6

Correct answer:

x=7,-6

Explanation:

To solve absolute value equations, you have to set up the equation two different ways to account for two answers.

The first is how you see it:

2x-1=13 .

The second is by making the answer negative:

2x-1=-13 .

Next solve each one for x. The first equation yields x to be 7. The second one gives you x=-6. Those are your two answers.

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Example Question #15 : Solving Absolute Value Equations

Solve for :

$\left | x-5\right |+8<3$

Possible Answers:

$x<-6 \and\ x>16$

$x<0 \and\ x>10$

No solution

x<0

Correct answer:

No solution

Explanation:

Given

$\left | x-5\right |+8<3$

first subtract 8 from both sides to get on the right.

$\left | x-5\right |<-5$

Based upon what we know about absolute value functions, it is impossible for an absolute value to be negative. So it is also impossible for an absolute value to be less than a negative number, which in this case is (-5) .

The only answer can be No Solution.

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Example Question #12 : Absolute Value

Solve for :

$2 \left | 3x-1 \right | = 10$

Possible Answers:

1.5; -1.5

2; -2

2; 1.5

$2; -\frac{4}{3}$

1;0

Correct answer:

$2; -\frac{4}{3}$

Explanation:

Solve using inverse opperations:

$2 \left | 3x-1 \right | = 10$

Divide both sides by 2.

|3x - 1 | = 5

Because the absolute value function always makes whatever is inside positive, this gives us two solutions. 3x - 1 could be either or , so we'll solve for both:

3 x - 1 = 5

3x = 6

x = 2

3x - 1 = -5

3x = -4

$x = \frac{ -4}{3}$

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Example Question #12 : Absolute Value

Solve for :

|2x - 7 | - 4 = 9

Possible Answers:

-1; 1

10; -10

10; 1

10; -3

-3; 1

Correct answer:

10; -3

Explanation:

In order to solve the equation, we have to isolate the variable. We do this by performing the same operation to either side of the equation.

|2x - 7 | - 4 = 9

First add 4 to both sides

|2x - 7 | = 13

The absolute value function makes whatever is inside positive, which means that there are two possible solutions. 2x - 7 could either be or
so we'll solve for both

2x - 7 = 13

2 x = 20

x = 10

2 x - 7 = -13

2x = -6

x = -3

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Example Question #21 : Absolute Value

Solve for :

$\frac{ |1-3x | }{4 } = 5$

Possible Answers:

$6.\overline{3}; - 7$

$\frac{ 19 }{ 3 }; -\frac{ 19 }{ 3 }$

8; -8

$-\frac{ 19 }{3} ; 7$

-7; -8

Correct answer:

$-\frac{ 19 }{3} ; 7$

Explanation:

In order to solve the equation, we have to isolate the variable. We do this by performing the same operation to either side of the equation.

$\frac{ |1-3x | }{4 } = 5$

First multiply both sides by 4.

| 1 - 3x | = 20

The absolute value function turns whatever is inside positive, so this has two solutions. 1 - 3 x could be positive or negative 20, so we'll solve for both:

1 - 3x = 20

Next subtract 1 from both sides.

-3x = 19

Then, divide by -3.

$x = \frac{ -19 }{3}$

1 - 3x = -20

Now subtract 1 from both sides.

-3x = -21

Finally, divide both sides by -3

x = 7

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Example Question #22 : Absolute Value

Solve for :

3|4x - 11 | = 27

Possible Answers:

$3.1\overline6; -3.1\overline6$

5; -5

$- \frac{ 1}{2} ; 5$

$3.1\overline{6}; -1.\overline{3}$

$5; \frac{ 1}{2}$

Correct answer:

$5; \frac{ 1}{2}$

Explanation:

In order to solve the equation, we have to isolate the variable. We do this by performing the same operation to either side of the equation.

3|4x - 11 | = 27

First, divide both sides by 3.

|4x - 11| = 9

The absolute value function makes whatever is inside positive, so this has two potential solutions. 4x - 11 could either be positive or negative 9, so we have to solve for both:

4 x - 11 = 9

4 x = 20

x = 5

4 x - 11 = - 9

4 x = 2

$x = \frac{ 1}{2}$

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Example Question #23 : Absolute Value

If g(x) = |x+3| , find g(-7) .

Possible Answers:

$-\frac{1}{4}$

$\frac{1}{4}$

Correct answer:

Explanation:

If g(x) = |x+3| , find g(-7) .

Substitute negative seven in for every x.

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

g(-7) = |-4|

Recall that the absolute value of a number always becomes positive.

g(-7) = 4

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Example Question #24 : Absolute Value

Solve the absolute value equation.

4 + |2x-1| = 7

Possible Answers:

x=2 and x=-1

x=-2 and x=1

x=-1

x=2

Correct answer:

x=2 and x=-1

Explanation:

4 + |2x-1| = 7

|2x-1| = 3

Since absolute values are concerened with the distance from zero, you need two equations to show both possible solutions.

FIRST

$\\2x-1=3\\ 2x=4\\ x=2$

SECOND

$\\2x-1=-3\\ 2x=-2\\ x=-1$

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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 #12 : Solving Absolute Value Equations

Example Question #13 : Solving Absolute Value Equations

Example Question #14 : Solving Absolute Value Equations

Example Question #15 : Solving Absolute Value Equations

Example Question #12 : Absolute Value

Example Question #12 : Absolute Value

Example Question #21 : Absolute Value

Example Question #22 : Absolute Value

Example Question #23 : Absolute Value

Example Question #24 : Absolute Value

All Algebra II Resources

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