lesson 24 equations with negatives - diablo valley...

Free Pre-Algebra Lesson 24 page 1

© 2010 Cheryl Wilcox

Lesson 24

Equations with Negatives

You’ve worked with equations for a while now, and including negative numbers doesn’t really change any of the rules. Everything you’ve already learned is still valid. But it’s still nice to go through some examples carefully and catch the little details that sometimes feel tricky in this new situation. Review About Equations Now that you’ve solved equations for a few weeks, let’s review exactly what we’re doing. With your added experience, you’ll have insights you perhaps didn’t the first time. An equation is a kind of mathematical sentence, and the equals sign is like the verb. In fact, when translating words to algebra, we usually translate the word “is” to an equals sign. An equals sign is a very, very strong symbol in mathematics. Many students think of it as a sort of “and next” symbol, since it often comes between the steps of a problem. But it is much stronger than that – it means that the things it is linking are identical and interchangeable, merely disguised versions of one another. Let’s look at an example to help a little with clarity in this abstract discussion. Here is an equation:

2x 1 7

The equals sign separates the two sides of the equation, and indicates that they are EQUAL, that is, they are exactly the same, mathematically speaking, because they represent the SAME AMOUNT. Here the two sides are 2x – 1 and 7.

To solve the equation means to find any numbers that can replace x to make the equation true, that is, to make the two sides equal. Those numbers are called solutions to the equation.

The procedure for finding solutions is to strip away the operations that were done to the variable by doing the opposite operations in the reverse order on both sides of the equals sign. Because the two sides begin equal and have exactly the same experiences, they stay equal to one another. The goal is to have the variable alone on one side of the equal sign (sometimes called isolating the variable) and any solutions on the other.

2x 1 7 2x 1 1 7 1

2x 8 2x / 2 8 / 2

x 4

The solution can replace x in the original equation and the two sides are equal:

x 4 2x 1 7

2(4) 1

8 1

7



Equations with Only One Operation We start with simple equations to see how to deal with negatives in each of the four arithmetic operations. Adding or Subtracting If a negative number is added to or subtracted from the variable, simplify first, then proceed as formerly.

Example: Solve the equation x + (–3) = 5.

First simplify by re-writing the equation as a subtraction: x – 3 = 5 Then solve as usual by doing the opposite operation on both sides.

x 3 5 x 3 3 5 3

x 8

Example: Solve the equation x – (–7) = 2.

First simplify by re-writing the equation as an addition: x + 7 = 2 Then solve as usual by doing the opposite operation on both sides. The solution to this equation is a negative number.

x 7 2 x 7 7 2 7

x 5

Multiplying or Dividing When undoing a multiplication or division with a negative number, you must do the opposite operation with the same negative number.

Example: Solve the equation –3x = 45.

In order to undo the operation of multiplying by –3 we must divide by –3, because –3 / (–3) = 1.

3x 45 3x / ( 3) 45 / ( 3)

x 15

Example: Solve the equation x /(-2) = 4.

To cancel the –2 in the denominator, both sides must be multiplied by –2. Another negative number solution.

x

24

x

2• ( 2) 4 • ( 2)

x 8

Multi-Operation Equations with Negatives This is just practice. Nothing is different, or new, now that you know how to deal with a negative number in each operation. Example: Solve the equations.

5x 9 39

5x 9 39 5x 9 9 39 9

5x 30 5x / ( 5) 30 / ( 5)

x 6

x 7

26

x 7

26

x 7

2•2 6 •2

x 7 12 x 7 7 12 7

x 5



Dealing with –x One way to deal with –x is to remember that –x is just shorthand for –1x. Example: Solve the equations.

x 5

1x 5 1x / ( 1) 5 / ( 1)

x 5

x 5

1x 5 1x / ( 1) 5 / ( 1)

x 5

Once you’ve gone through these steps a few times it begins to feel like overkill. A slightly shorter thought process occurs if you interpret –x as “the opposite of x.” Then you can “take the opposite” of both sides of the equation without writing a division.

You should use whichever of these method makes the most sense to you, or any other method that you understand and that works. (Some people multiply both sides of the equation by –1, which has the same result.) Just remember that if your solution says –x instead of x, you’re not quite done with the problem yet, and have one more simplification step.

Example: Solve the equation 9 – x = 2.

9 x 2

It’s important to remember that the subtraction sign is applied to the x, not the 9. If we re-write the equation to change the subtraction to addition, you can see it more clearly. Each of the gray

equations to the right is equivalent to the equation above.

9 x 2

9 1x 2

1x 9 2

The last version certainly looks like something we already know how to solve.

1x 9 2 1x 9 9 2 9

1x 7 1x / ( 1) 2 / ( 1)

x 7

If we want to solve the original equation without rearranging and using the “take the opposite” shortcut, it looks like this:

9 x 2 9 x 9 2 9

x 7

x 7

The tricky part is to subtract 9 from both sides, because the 9 is a positive number added to –x. Many people see the – sign and think they need to add 9, but the minus applies to the x.

Check: x 7 9 x 2

9 (7)

2



Example: Solve the equation –6 – x = 2.

Rewrite the problem by changing the subtraction to + –x or to + –1x.

Option 1

6 x 2

6 x 2

x 6 2

x 6 2

Option 2

6 x 2

6 1x 2

1x 6 2

1x 6 2

Then solve the equation in its new form.

Option 1

x 6 2 x 6 6 2 6

x 8

x 8

Option 2

1x 6 2 1x 6 6 2 6

1x 8 1x / ( 1) 8 / ( 1)

x 8

Check: x 8

6 x 2

6 ( 8)

6 8

2 true



Negative Cheat Sheet

Type 1 Models

Type 2 Models

Comparing with > and <

100 1 Any negative number is less than any positive number.

100 99 It is worse to owe $100 than to owe $99, or, –100º is colder than –99º, etc.

Absolute Value:

makes everything positive:

5 5 5 . Also, 0 0

gives priority like parentheses in the order of operations:

2 5 6 2 • 1 2 •1 2

Addition Battle or Party?

Battles

5 3 2 3 5 2

Parties

3 5 8 3 5 8

Subtraction Change to addition.

3 5

3 5 2

3 ( 5)

3 5 8

Multiplication & Division Like signs, positive, unlike signs, negative.

3 • 5 15

15 / 5 3

3 • 5 15

15 / 5 3

3 • 5 15

15 / 5 3

3 • 5 15

15 / 5 3

Be careful with subtle notation differences: compare the multiplication 3( 5) to the subtraction 3 ( 5)

Exponents Write out the multiplication.

( 3)2 ( 3)( 3) 9 32 (32 ) (3 • 3) 9

( 3)3 ( 3)( 3)( 3) 27 33 (33 ) (3 • 3 • 3) 27

Fractions are divisions.

8

2

( 8)

2

8

( 2)4

( 8)

( 2)4

2

3

2

3

2

3

2

3

2

3

Algebraic Expressions have super-efficient notation.

3x 5x 2x 3x 4x 1x x 3x 3x 0x 0 3x 2x 1x x

3(2x 1) 6x ( 3) 6x 3 3(2x 1) 3(2x ( 1)) 6x 3

Formulas Parentheses around numbers when substituting!

t 3 h 16t 2 80t 64

h 16(3)2 80(3) 64 160

x 3 x 2 x

( 3)2 ( 3) 9 3 12

Interpret negative answers in the context of the problem (loss vs. profit, below vs. above ground, etc.)

Equations It can help to convert –x to –1x before solving.

If x 5, then x 5.

If x 5, then x 5.

To solve 3 x 4 change to

1x 3 4 1x 3 3 4 3

1x 7 1x / ( 1) 7 / ( 1)

x 7



Lesson 24: Equations with Negatives

Worksheet Name ________________________________________

Solve the equations. The answers may be positive or negative whole numbers or fractions.

8a 5 13

2b 7 19

21 3(c 9)

14 d 45

k 7

21

3m

57

4 3x 2x 18 2

x

22 16

Choose one of the problems above and check the answer in the original equation.




Homework 24A Name ______________________________

1. Translate the words to mathmatical notation.

a. It is better to owe $20 than to owe $100.

b. The product of –1 and 4 is –4.

c. When 5 positive particles are combined with 4 negative particles the result is that 1 positive particle remains.

2. Find the absolute value:

a.

5

b. 0

c. 9

d.

( 1)

3. Compare with > or <.

a. 7 0

b. 7 3

c. 7 10

d. 7 10

4. Compare with <, >, or =.

a. 7 1 1 7

b. (7 1) (1 7)

c. (7)( 1) ( 1)( 7)

5. Simplify.

9 8

9( 8)

9 ( 8)

9 ( 8)

9 / ( 1)

9 ( 1)

52

( 5)2

( 1)( 2)( 3)

1 9

4

2 6 ( 5)

9 4 1 10

6. Simplify the algebraic expressions. Use super-efficient notation in your answers.

4x 8 5x

9 6x ( 5x)

9x 8y 7x 2(4y)



7. Use the given formula to solve the problem.

The height above ground in feet of a small model rocket fired from ground level (disregarding air resistance) is given by the equation

h 16t2 192t

The rocket reaches its greatest height after 6 seconds. What is the height when t = 6?

What is the height when t = 12?

Convert –200ºC to Fahrenheit using the formula

F

9C

532

Convert –193ºF to Celsius using the formula

C

5(F 32)

9

8. Evaluate each expression for the given values of the variable.

x2 2x

x x2

9. Solve the equations.

4x 9 3

6 x 7




Homework 24A Answers


a. It is better to owe $20 than to owe $100.

20 100

b. The product of –1 and 4 is –4.

( 1)(4) 4

c. When 5 positive particles are combined with 4 negative particles the result is that 1 positive particle remains.

5 4 1


a.

5 5

b. 0 0

c. 9 9

d.

( 1) 1


a. 7 < 0

b. 7 < 3

c. 7 > 10

d. 7 < 10


a. 7 1 1 7

6 6

b. (7 1) (1 7)

6 6

c.

(7)( 1) ( 1)( 7)

7 7

5. Simplify.

9 8

9 8 17

9( 8)

72

9 ( 8)

9 8 1

9 ( 8)

9 8 17

9 / ( 1)

9

9 ( 1)

8

52

(5 • 5) 25

( 5)2

( 5)( 5) 25

( 1)( 2)( 3)

(2)( 3)

6

1 9

4

8

4

8

4

2

2 6 ( 5)

2 6 5 2 11

2 • 11 22

9 4 1 10

(9 1) 10 4

(10 10) 4

4


4x 8 5x

4x 5x 8

1x 8

x 8

9 6x ( 5x)

9 6x 5x

9 1x

9 x

9x 8y 7x 2(4y)

9x 7x 8y 8y

2x 0y

2x





h 16t2 192t


h 16(6)2 192(6)

16(36) 192(6)

576 1152

576 feet above ground


h 16(12)2 192(12)

16(144) 192(12)

2304 2304 0

The rocket hits ground.


F

9C

532

C 200

F9( 200

40

)

532 360 32 328ºF


C

5(F 32)

9

F 193

C5(( 193) 32)

9

5( 225

25

)

9125ºC


x2 2x

x x2


4x 9 3

4x 9 3 4x 9 9 3 9

4x 6 4x / 4 6 / 4

x6

4

3

2

6 x 7

x 6 7 x 6 6 3 6

4x 6 4x / 4 6 / 4

x6

4

3

2




Homework 24B Name ________________________________


a. Owing $75 is better than owing $100.

b. The quotient of 4 and –4 is –1.

c. The checking account had $30, but there was a debit of $50, resulting in an overdraft of $20.


a. 0

b.

1

c. 3

d. 0 5


a. 9 7

b. 9 10

c. 9 9

d. 9 90


a. 9 1 9 ( 1)

b. 9 1 1 ( 9)

c.

9 1 1 ( 9)

5. Simplify.

4 2

4 2

4( 2)

4 / ( 2)

4 2

4 ( 2)

( 4)2

42

( 2)( 3)( 4)

6 8 3

3

6 2 • 3

32

( 3)2


3x 7x 4x

6y 2 5y 3

3(x 2)





h 16t2 224t




F

9C

532


C

5(F 32)

9


3 3x

3x2 3


8y 6 5

1 7 x

lesson 24 equations with negatives - diablo valley...

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