Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #63 : Solving Absolute Value Equations

Solve:  \displaystyle 3\left | x-1\right |+7 =6

Possible Answers:

\displaystyle -\frac{2}{3}, \frac{4}{3}

\displaystyle \textup{No solution.}

\displaystyle -\frac{3}{4}, \frac{2}{3}

\displaystyle \frac{3}{4}, 2

\displaystyle -\frac{2}{3}, \frac{2}{3}

Correct answer:

\displaystyle \textup{No solution.}

Explanation:

Start by isolating the absolute value.

Subtract seven on both sides.

\displaystyle 3\left | x-1\right |+7 -7=6-7

\displaystyle 3\left | x-1\right |=-1

Divide by three on both sides.

\displaystyle \frac{3\left | x-1\right |}{3}=\frac{-1}{3}

\displaystyle \left | x-1\right |=-\frac{1}{3}

There is no value of \displaystyle x in an absolute value that will give a negative value since the absolute value will convert all numbers to positive.

The answer is:  \displaystyle \textup{No solution.}

Example Question #64 : Solving Absolute Value Equations

Solve for x: \displaystyle |3-x| = 19

Possible Answers:

\displaystyle x = -16, 22

\displaystyle x=-16, 16

\displaystyle x= 21, 16

\displaystyle x=-21, 21

Correct answer:

\displaystyle x = -16, 22

Explanation:

What is inside the absolute value brackets could be positive or negative.

Positive: \displaystyle 3-x = 19

\displaystyle -x = 16

\displaystyle x = -16

Negative: \displaystyle 3-x = -19

\displaystyle -x = -22

\displaystyle x= 22

Example Question #65 : Solving Absolute Value Equations

Solve:  \displaystyle \left | 8-2x\right |= 3x-2

Possible Answers:

\displaystyle -6, 2

\displaystyle -2,6

\displaystyle \textup{There is no solution.}

\displaystyle -6, -2

\displaystyle 2

Correct answer:

\displaystyle 2

Explanation:

Separate the absolute value and solve both the positive and negative components of the absolute value.

\displaystyle 8-2x = 3x-2

\displaystyle -(8-2x) = 3x-2

Solve the first equation.  Add \displaystyle 2x on both sides.

\displaystyle 8-2x +2x= 3x-2+2x

\displaystyle 8= 5x-2

Add two on both sides.

\displaystyle 10=5x

Divide by five on both sides.

\displaystyle \frac{10}{5}=\frac{5x}{5}

One of the solutions is \displaystyle x=2 after substitution is valid.

Evaluate the second equation.  Distribute the negative sign in the binomial.

\displaystyle -8+2x= 3x-2

Subtract \displaystyle 2x on both sides.

\displaystyle -8+2x-2x= 3x-2-2x

\displaystyle -8 = x-2

Add two on both sides.

\displaystyle -8+2 = x-2+2

\displaystyle x=-6

If we substitute this value back to the original equation, the equation becomes invalid.  

The answer is:  \displaystyle 2

Example Question #61 : Absolute Value

Solve the absolute value equation:  \displaystyle 2\left | 3x-3\right | =16

Possible Answers:

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

\displaystyle \textup{There is no solution.}

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

\displaystyle \frac{5}{3}, \frac{11}{3}

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

Correct answer:

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

Explanation:

Divide by two on both sides to isolate the absolute value.

\displaystyle \frac{2\left | 3x-3\right | }{2}=\frac{16}{2}

The equation becomes:

\displaystyle \left | 3x-3\right | = 8

Split this equation into its positive and negative components.

\displaystyle 3x-3 = 8

\displaystyle -(3x-3)=8

Solve the first equation.  Add three on both sides.

\displaystyle 3x-3+3 = 8+3

\displaystyle 3x=11

Divide by three on both sides.

\displaystyle \frac{3x}{3}=\frac{11}{3}

The first solution is:  \displaystyle \frac{11}{3}

Evaluate the second equation.  Divide by negative one on both sides.

\displaystyle \frac{-(3x-3)}{-1}=\frac{8}{-1}

\displaystyle 3x-3=-8

Add three on both sides.

\displaystyle 3x-3+3=-8+3

\displaystyle 3x=-5

Divide by three on both sides.

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

The second solution is:  \displaystyle -\frac{5}{3}

The answers are:  \displaystyle -\frac{5}{3}, \frac{11}{3}

Example Question #61 : Absolute Value

Evaluate:  \displaystyle -3\left | 9x+2\right | = 33

Possible Answers:

\displaystyle \frac{13}{9},-1

\displaystyle -\frac{13}{9},-1

\displaystyle \frac{13}{9},1

\displaystyle \textup{No solution.}

\displaystyle -\frac{13}{9},1

Correct answer:

\displaystyle \textup{No solution.}

Explanation:

Isolate the absolute value by dividing both sides by negative three.

\displaystyle \frac{-3\left | 9x+2\right | }{-3}=\frac{ 33}{-3}

The equation becomes:  

\displaystyle \left | 9x+2\right | = -11

Recall that an absolute value cannot be negative since the absolute value will change negative values to positive.

The answer is:  \displaystyle \textup{No solution.}

Example Question #68 : Solving Absolute Value Equations

Solve:  \displaystyle \left | -3x+2\right |+3x = 6

Possible Answers:

\displaystyle -\frac{1}{3}, \frac{4}{3}

\displaystyle \textup{There is no solution.}

\displaystyle -\frac{3}{4}

\displaystyle \frac{4}{3}

\displaystyle \frac{1}{3}, \frac{4}{3}

Correct answer:

\displaystyle \frac{4}{3}

Explanation:

Subtract \displaystyle 3x on both sides.

\displaystyle \left | -3x+2\right |+3x -3x= 6-3x

\displaystyle \left | -3x+2\right | = -3x+6

Evaluate the positive and negative components of the absolute value.  The two equations are:

\displaystyle -3x+2 = -3x+6

\displaystyle -(-3x+2 )= -3x+6

Evaluate the first equation.  Add \displaystyle 3x on both sides.

\displaystyle -3x+2 +3x= -3x+6+3x

\displaystyle 2=6

There is no solution for the first equation.

Evaluate the second equation.  Simplify the left side by distributing negative one through the binomial.

\displaystyle 3x-2 = -3x+6

Add \displaystyle 3x on both sides.

\displaystyle 3x-2 +3x= -3x+6+3x

The equation becomes:

\displaystyle 6x-2 = 6

Add two on both sides.

\displaystyle 6x-2 +2= 6+2

\displaystyle 6x=8

Divide by six on both sides and reduce the fraction.  Verify that the answer will satisfy the original equation.

\displaystyle \frac{6x}{6}=\frac{8}{6} = \frac{4}{3}

The answer is:  \displaystyle \frac{4}{3}

Example Question #62 : Absolute Value

Solve the absolute value equation:  \displaystyle -2\left |\frac{1}{3}x-7 \right |-6 = -5

Possible Answers:

\displaystyle \pm\frac{39}{2}

\displaystyle \frac{39}{2}

\displaystyle -\frac{45}{2},\frac{39}{2}

\displaystyle \textup{No solution.}

\displaystyle -\frac{45}{2}

Correct answer:

\displaystyle \textup{No solution.}

Explanation:

Evaluate by adding six on both sides.

\displaystyle -2\left |\frac{1}{3}x-7 \right |-6+6 = -5+6

\displaystyle -2\left |\frac{1}{3}x-7 \right |=1

Divide by negative two on both sides.

\displaystyle \frac{-2\left |\frac{1}{3}x-7 \right |}{-2}=\frac{1}{-2}

The equation becomes:  \displaystyle \left |\frac{1}{3}x-7 \right | = -\frac{1}{2}

Recall that absolute values will convert all negative values to positive.  No matter what \displaystyle x would evaluate to be, it will never become negative half.

The answer is:   \displaystyle \textup{No solution.}

Example Question #70 : Solving Absolute Value Equations

Solve:  \displaystyle 8\left |x-3 \right | = 7

Possible Answers:

\displaystyle - \frac{31}{8},-\frac{17}{8}

\displaystyle \textup{No solution.}

\displaystyle -\frac{17}{8}, \frac{31}{8}

\displaystyle - \frac{31}{8},\frac{17}{8}

\displaystyle \frac{17}{8}, \frac{31}{8}

Correct answer:

\displaystyle \frac{17}{8}, \frac{31}{8}

Explanation:

Divide both sides by eight.

\displaystyle \frac{8\left |x-3 \right | }{8}= \frac{7}{8}

\displaystyle \left |x-3 \right | = \frac{7}{8}

Split the absolute value into its positive and negative components.

\displaystyle x-3 = \frac{7}{8}

\displaystyle -(x-3 )= \frac{7}{8}

Solve the first equation.  Multiply both sides by eight to eliminate the fraction.

\displaystyle 8(x-3) = \frac{7}{8}\cdot 8

\displaystyle 8(x-3) = 7

Use distribution to simplify.

\displaystyle 8x-24 = 7

Add 24 on both sides, and then divide both sides by eight.

\displaystyle 8x-24 +24= 7+24

\displaystyle 8x=31

\displaystyle \frac{8x}{8}=\frac{31}{8}

The first solution is \displaystyle \frac{31}{8}.

Solve the second equation.  Divide by negative one on both sides.

\displaystyle \frac{-(x-3 )}{-1}= \frac{\frac{7}{8}}{-1}

The equation becomes:

\displaystyle x-3 = -\frac{7}{8}

Add three on both sides.  This is the same as adding \displaystyle \frac{24}{8} on both sides.

\displaystyle x-3+(3) = -\frac{7}{8}+(\frac{24}{8})

\displaystyle x=\frac{17}{8}

The answers are:  \displaystyle \frac{17}{8}, \frac{31}{8}

Example Question #71 : Solving Absolute Value Equations

Solve.

\displaystyle 6-\left | 2(x+4)\right |=0

Possible Answers:

\displaystyle x=7 or \displaystyle x=1

\displaystyle x=-7 or \displaystyle x=-1

\displaystyle x=-7 or \displaystyle x=1

\displaystyle x=-7 and \displaystyle x=-1

\displaystyle x=7 and \displaystyle x=1

Correct answer:

\displaystyle x=-7 or \displaystyle x=-1

Explanation:

Solve.

\displaystyle 6-\left |2(x+4) \right |=0

Step 1: Isolate the absolute value by subtracting \displaystyle 6 from both sides of the equation.

\displaystyle -\left | 2(x+4)\right |=-6

Step 2: Divide -1 from both sides of the equation in order to get rid of the negative sign in front of the absolute value.

\displaystyle \left | 2(x+4)\right |=6

Step 3: Because this is an inequality, this equation can be solved in two parts as shown below.

Note:  \displaystyle \left | f(x)\right |=a can be written as \displaystyle f(x)=-a or \displaystyle f(x)=a

\displaystyle 2(x+4)=-6  or  \displaystyle 2(x+4)=6

Step 4: Solve both parts.

\displaystyle 2(x+4)=-6

Distribute the \displaystyle 2.

\displaystyle 2x+8=-6

Subtract \displaystyle 8 from both sides of the equation.

\displaystyle 2x=-14

Divide both sides of the equation by \displaystyle 2.

\displaystyle x=-7

\displaystyle 2(x+4)=6

Distribute the \displaystyle 2.

\displaystyle 2x+8=6

Subtract \displaystyle 8 from both sides of the equation.

\displaystyle 2x=-2

Divide both sides of the equation by \displaystyle 2.

\displaystyle x=-1

Step 5: Combine both parts using "or".

\displaystyle x=-7 or \displaystyle x=-1

Solution: \displaystyle x=-7 or \displaystyle x=-1

Example Question #72 : Solving Absolute Value Equations

Solve for\displaystyle x values given the equation \displaystyle |10x+7|=50

Possible Answers:

\displaystyle 43/10 and \displaystyle 57/10

\displaystyle -43/10 and \displaystyle -57/10

\displaystyle 43/10 and  \displaystyle -57/10

\displaystyle -57/10

\displaystyle 43/10

Correct answer:

\displaystyle 43/10 and  \displaystyle -57/10

Explanation:

Given: \displaystyle |10x+7|=50 

When given an absolute value recognize there are often multiple solutions. The reason why is best exhibited in a simpler example:

Given \displaystyle |x|=1 solve for values \displaystyle X of that make this statement true. When you taken an absolute value of something you always end up with the positive number so both \displaystyle -1 and \displaystyle 1 would make this statement true. The solutions can also be written as \displaystyle ±1.

In the case of the more complicated equation \displaystyle |10x+7|=50 for the same reason there are potentially two solutions, which are shown by \displaystyle ±(10x+7)=50 as an absolute value will always end up creating a positive result.

To simplify the absolute value we must look at each of these cases:

\displaystyle 1) Let's start with the positive case:

\displaystyle 10x+7=50

Just like a normal equation with one unknown we will simplify it by isolating \displaystyle x by itself. This is first done by subtracting \displaystyle 7 from both sides leaving:
\displaystyle 10x=43

Next \displaystyle 10 is divided from both sides leaving \displaystyle x=43/10, as your final solution.

To check this solution it must be substituted in the original absolute value for \displaystyle x and if it's a correct answer you'll end up with a true statement, so

\displaystyle |10x+7|=50\displaystyle x=43/10

\displaystyle |10(43/10)+7|\displaystyle *43/10*10=43

so this becomes:

\displaystyle |43+7|=50

and the absolute value of \displaystyle 50 is \displaystyle 50 so you end up with a true statement. Therefore \displaystyle x=43/10  is a valid solution

\displaystyle 2) Next let's solve for the negative case:

\displaystyle -(10x+7)=50

Distribute the negative sign, which is just \displaystyle -1 to make calculations easier and you'll get:

\displaystyle -10x-7=50

Next \displaystyle 7 can be added to both sides, giving

\displaystyle -10x=57

dividing by \displaystyle -10 leaves:

\displaystyle x=-57/10

 Checking this solution is done just as you did for the previous solution obtained.

Given \displaystyle |10x+7|=50\displaystyle x=-57/10

substitute \displaystyle -57/10  in for \displaystyle x

\displaystyle |10(-57/10)+7|=50

multiply \displaystyle 10*(-57/10) gives \displaystyle -57

so you obtain \displaystyle |-57+7|=50

adding \displaystyle -57+7=-50

and the absolute value of \displaystyle -50 is \displaystyle 50 thereby making this also a valid solution, therefore the two valid solutions are  \displaystyle x=43/10 and \displaystyle -57/10

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