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The key to solving any equation is getting the variable all alone on one side of the equation. You isolate the variable by performing inverse operations. If the variable has been multiplied by a number, you solve the equation by dividing both sides by that number.

If the variable has been divided by a number, you multiply both sides by that number to find the value of the variable.

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EXERCISE 2.2

Solve each equation by multiplying or dividing both sides by the appropriate number.

Two-step equations

Most equations require two or more operations to find a solution. The equation 4x − 5 = 19 says that if you start with a number x, multiply it by 4, and then subtract 5, the result is 19. To solve for x, you will need to perform the opposite, or inverse, operations in the opposite order. You are undoing, stripping away, what was done to x and working your way back to where things started. Undo the subtraction by adding 5.

4x − 5 = 19

4x = 19 + 5

4x = 24

Then undo the multiplication by dividing by 4.

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EXERCISE 2.3

Solve each equation.

1. 3x − 7 = 32

2. −5t + 9 = 24

3. 4 − 3x = −11(Rewrite as −3x + 4 = −11 if that’s easier.)

4. 9 + 3x = 10

6. −3x + 5 = −16

8. 11 − 3x = 9.5

10. 2x − 7 = −23

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Variables on both sides

When variable terms appear on both sides of the equation, add or subtract to eliminate one of them. This should leave a one- or two-step equation for you to solve.

3x−2x − 7 = 2x−2x + 4

x − 7 = 4

x = 11

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EXERCISE 2.4

Eliminate the extra variable term by adding or subtracting; then solve the equation.

1. 5x − 8 = x + 12

2. 11x + 18 = 3x − 14

3. 3x + 8 = 4x − 9

4. 9 − 4x = 16 + 3x

5. 2x − 5 = 3 − 4x

6. 8x − 17 = 12 + 3x

7. x − 3 = 2 − x

8. 1.5x − 7.1 = 8.4 + x

9. 7 − 9x = 7x − 19

10. −5x + 21 = 27 − x

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Simplifying before solving

If the equation contains parentheses or has more than two terms on either side, take the time to simplify each side of the equation before you try to solve. If there is a multiplier in front of the parentheses, use the distributive property to multiply and remove the parentheses. Focus on one side at a time and combine like terms. There should be no more than one variable term and one constant term on each side of the equation when you start the process of solving by inverse operations.

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EXERCISE 2.5

Simplify the left side and the right side of each equation. Leave no more than one variable term and one constant term on each side. Then solve each equation.

1. 5(x + 2) = 40

2. 4(x − 7) + 6 = 18

3. 5(x − 4) = 7(x − 6)

4. 4(5x + 3) + x = 6(x + 2)

5. 8(x − 4) − 16 = 10(x − 7)

6. 6(2x + 9) − 30 = 4(7x − 2)

7. 7(x − 1) + 2x = 12 + 5(x + 1)

8. 6(x − 1) −2x = 2(x + 1) + 4(2 − x)

9. 5(6x + 2) + 7(4 − 12x) = 35 −(6 + 27x)

10. 8(2x − 5) − 2(x − 2) = 5(x + 7) − 4(x + 8)

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Absolute value equations

An absolute value equation, with a variable expression inside the absolute value signs, isn’t actually a linear equation, but it’s closely related. The graphs of absolute value functions aren’t lines; they’re V-shaped. Absolute value equations will usually have two solutions. The expression between the absolute value signs may be equal to the expression on the other side of the equal sign, or it may be equal to the opposite of that expression.

Always isolate the absolute value before considering the two cases. All terms other than the absolute value should be moved to the other side before you say that the expression in the absolute value signs could equal the other side or its opposite. If there are two or more terms on that side, be sure to form the opposite correctly by distributing the negative.

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EXERCISE 2.6

Each equation involves absolute value. Solve each equation by the method above.

1. |3x + 5| = 23

2. |6x − 3| = 17

3. |5x + 2| = 47

4. |3 + 6x| = 33

5. |7 + 8x| = 51

6. |30 + 3x| = 18x

7. |40 − 2x| = 6x

8. |3x − 11| = 8 + x

9. |8x − 2| = 2x + 22

10. |9x + 2| − 3x = 17 + x

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Mixture problems: coffee,