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Algebra 1 : Linear Equations

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #16 : How To Solve Absolute Value Equations

Solve for :

| 2x + 15 | = -77

Possible Answers:

x = 31

$x \in \left \{ 31,46\right \}$

x = 31

There is no solution.

$x \in \left \{ -31,46\right \}$

$x \in \left \{ -46, 31\right \}$

Correct answer:

There is no solution.

Explanation:

The absolute value of a number can never be a negative number. Therefore, no value of can make | -3x - 15 | = -42 a true statement.

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Example Question #12 : How To Solve Absolute Value Equations

Solve for .

|x+3|=1

Possible Answers:

x=-4

$x=0\ \text{or}\ x=-3$

$x=-2\ \text{or}\ x=2$

$x=0\ \text{or}\ x=3$

$x=-2\ \text{or}\ x=-4$

Correct answer:

$x=-2\ \text{or}\ x=-4$

Explanation:

|x+3|=1

The equation involves an absolute value. First, we need to rewrite the equation with no absolute value.

$x+3=\pm 1$

We can split this equation into two possible equations.

Equation 1: x+3 =1

Equation 2: x+3 =-1

With two equations, there are two values for . Let's start with Equation 1.

x+3 =1

Subtract from both sides.

x+3-3=1-3

x=-2

That's the first value for . To get the second value for , we need to repeat the steps, but with Equation 2.

x+3 =-1

x+3-3=-1-3

x=-4

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Example Question #851 : Algebra 1

Solve for :

$\left | 2x-7 \right | -15 = -6$

Possible Answers:

$x=-8\textup{ or }x=8$

$x=-1\ \textup{or }x=8$

$x=-8\ \textup{or }x=1$

x=8

$x=-1\textup{ or }x=1$

Correct answer:

$x=-1\ \textup{or }x=8$

Explanation:

First, isolate the absolute value expression on one side.

$\left | 2x-7 \right | -15 = -6$

$\left | 2x-7 \right | -15 + 15= -6 + 15$

$\left | 2x-7 \right | =9$

Rewrite this as the compound statement:

2x-7 =-9 or 2x-7 =9

Solve each equation separately:

2x-7 =-9

2x-7+ 7 =-9 + 7

2x = -2

$2x\div 2 = -2 \div 2$

x=-1

2x-7 =9

2x-7+ 7 =9 + 7

2x = 16

$2x\div 2 = 16 \div 2$

x=8

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Example Question #18 : How To Solve Absolute Value Equations

Solve for :

$\left | 2x +1 \right |=7$

Possible Answers:

x=3, -3

x=4, -3

x=-4, 3

$x=\pm4$

Correct answer:

x=-4, 3

Explanation:

Absolute value is a function that turns whatever is inside of it positive. This means that what's inside the function, 2x+1 , might be 7, or it could have also been -7. We have to solve for both situations.

a. 2x+1=7 subtract 1 from both sides

2x=6 divide both sides by 2

x=3

b. 2x+1=-7 subtract 1 from both sides

2x=-8 divide both sides by 2

x=-4

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Example Question #21 : How To Solve Absolute Value Equations

$3(x+1)^2=\left | x+1\right |. \textup{Which is not a solution?}$

Possible Answers:

$-\frac{5}{3}$

$\textup{None of the above.}$

$-\frac{2}{3}$

$-\frac{4}{3}$

Correct answer:

$-\frac{5}{3}$

Explanation:

$\\ \textup{We get rid of the absolute value by solving the two equations }\\3(x+1)^2=x+1\\\textup{and}\\3(x+1)^2=-(x+1)\\\textup{If } x\neq1\textup{ we can divide in both equations the two sides by }\\ x+1 \textup{ and obtain }\\ 3(x+1)=1 \\\textup{ and }}\\ 3(x+1)=-1 \\\textup{ which give}}\\ \textup{the solutions } x=-\frac{2}{3} \textup{ and } x=-\frac{4}{3}.$

$\textup{If } x=-1, \textup{both sides are equal to 0, so } -1 \textup{ is another solution.}$

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Example Question #22 : How To Solve Absolute Value Equations

Solve for all possible values of .

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

Possible Answers:

x=-2,x=2/3

x=3,x=-2

x=2/3,x=-3

x=2,x=-3

x=-2/3, x=2

Correct answer:

x=-2/3, x=2

Explanation:

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

First, ignore absolute value signs and solve for x.

3x-2=-4

3x=-2

x=-2/3

Now, for the other solution, ignore absolute value signs and change the sign of the right side of the equation. Solve for x.

3x-2=4

3x=6

x=2

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Example Question #23 : How To Solve Absolute Value Equations

Solve for all values of .

$\left | 4x\right |-2=10$

Possible Answers:

x=3,x=2

x=-2,x=3

x=2,x=-2

x=3,x=-3

x=0,x=3

Correct answer:

x=3,x=-3

Explanation:

First, make sure to have the absolute value part of the equation by itself on the left side.

$\left | 4x\right |-2=10$

$\left | 4x\right |=12$

Now solve for both x values.

4x=12

x=3

Change the side of the right hand side to find the other value of x.

4x=-12

x=-3

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Example Question #24 : How To Solve Absolute Value Equations

Solve for .

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

Possible Answers:

x = 0,3

x = 0, -3

x = 3, -3

x = 1, -3

x = 0, 1

Correct answer:

x = 0,3

Explanation:

The question is asking for the value of . To isolate , we must first isolate the absolute value expression.

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

The absolute value bars give a positive value to the result of the expression within. This means the real value of the inner expression could be positive or negative. To find the two values of , create the two possible equations.

2x-3=3 $\boldsymbol{\mathbf{OR}}$ 2x-3=-3

Balance the equations to find .

2x-3=3

2x=6

x=3

2x-3=-3

2x=0

x=0

Therefore, x= 0 and x=3

Also written as x= 0,3 .

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Example Question #24 : How To Solve Absolute Value Equations

Solve for .

$\left | x\right |=10$

Possible Answers:

$\pm1$

$\pm10$

Correct answer:

$\pm10$

Explanation:

When solving with absolute values, we need to consider both positive and negative answers.

x=10

-x=10

Divide both sides by , we get x=-10 .

Final answer is $\pm10$ .

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Example Question #25 : How To Solve Absolute Value Equations

Solve for .

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

Possible Answers:

-3, -7

3, 7

Correct answer:

-3, -7

Explanation:

When solving with absolute values, we need to consider both positive and negative answers.

x+5=2

Subtract both sides by , we get x=-3 .

-(x+5)=2

Distribute the negative sign to get -x-5=2 .

Add both sides by and divide both sides by to get x=-7 .

Final answer is -3, -7 .

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Algebra 1 : Linear Equations

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #16 : How To Solve Absolute Value Equations

Example Question #12 : How To Solve Absolute Value Equations

Example Question #851 : Algebra 1

Example Question #18 : How To Solve Absolute Value Equations

Example Question #21 : How To Solve Absolute Value Equations

Example Question #22 : How To Solve Absolute Value Equations

Example Question #23 : How To Solve Absolute Value Equations

Example Question #24 : How To Solve Absolute Value Equations

Example Question #24 : How To Solve Absolute Value Equations

Example Question #25 : How To Solve Absolute Value Equations

All Algebra 1 Resources

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