Free Pre-Algebra
Lesson 24  page 1
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 8
x
2x 1 1 7 1
2x / 2 8 / 2
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
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 24  page 2
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 8
x 3 3 5 3
x 7 2
x
5
x 7 7 2 7
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.
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.
3x
In order to undo the operation of multiplying by –3 we must divide by –3,
because –3 / (–3) = 1.
x
Example: Solve the equation x /(-2) = 4.
x
To cancel the –2 in the denominator, both sides must be multiplied by –2.
2
4
x
Another negative number solution.
3x / ( 3)
45
15
x
2
• ( 2)
45 / ( 3)
4 • ( 2)
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.
x 7
5x 9 39
5x
9 39
5x 30
x
6
© 2010 Cheryl Wilcox
5x 9 9 39 9
5x / ( 5) 30 / ( 5)
6
2
x 7
2
x 7
x
6
12
5
x 7
2
•2
x 7 7
6•2
12 7
Free Pre-Algebra
Lesson 24  page 3
Dealing with –x
One way to deal with –x is to remember that –x is just shorthand for –1x.
Example: Solve the equations.
x
x
5
1x
x
5
5
1x / ( 1)
5
1x
5 / ( 1)
x
5
5
1x / ( 1)
5 / ( 1)
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
9
1x
x
1x
2
2
9 2
The last version certainly looks like something we already know how to solve.
1x
1x 9 9 2 9
1x / ( 1) 2 / ( 1)
9 2
1x
7
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
x
x
9 x
2
7
9 2 9
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
© 2010 Cheryl Wilcox
7
9 x 2
9 (7)
2
Free Pre-Algebra
Lesson 24  page 4
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
Option 2
6 x
2
1x
2
6
1x
1x
x 6 2
6 2
6 2
Then solve the equation in its new form.
Option 1
Check: x
x 6 2
x 8
x
8
8
6 x
x 6 6 2 6
Option 2
6
1x
x
2
6 ( 8)
6 8
2 true

© 2010 Cheryl Wilcox
1x
2
8
8
1x 6 6 2 6
1x / ( 1) 8 / ( 1)
Free Pre-Algebra
Lesson 24  page 5
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,
Absolute Value:
or, –100º is colder than –99º, etc.
5
makes everything positive:
5 . Also, 0
5
gives priority like parentheses in the order of operations:
25 6
Addition
0
Battles
2• 1
2•1 2
Parties
Battle or Party?
5 3
Subtraction
3 5
2
3 5 2
3
5
8
3 5 8
3•5
15
3• 5
15 / 5
3
3 ( 5)
Change to addition.
3
Multiplication
& Division
3 • 5 15
5
2
3 5 8
3 • 5 15
15 / 5 3
15 / 5
3
15
15 / 5 3
Like signs, positive,
unlike signs, negative.
Be careful with subtle notation differences: compare the multiplication 3( 5) to the subtraction 3 ( 5)
Exponents
( 3)2
Write out the multiplication.
( 3)
3
( 3)( 3) 9
32
( 3)( 3)( 3)
3
Fractions
are divisions.
8
2
Algebraic Expressions
3x 5x 2x
have super-efficient
notation.
Formulas
Parentheses around
numbers when substituting!
( 8)
2
3(2x 1)
t 3
8
( 2)
27
3
( 8)
( 2)
4
(3 • 3)
3
(3 )
9
(3 • 3• 3)
2
3
4
3x 4x 1x x
6x ( 3)
(32 )
2
3
3(2x 1)
16t 2 80t 64
h
16(3)2 80(3) 64 160
2
3
3x 3x 0x 0
6x 3
h
27
x
3
2
3
3x 2x
3(2x ( 1))
2
3
1x
x
6x 3
x2 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.
© 2010 Cheryl Wilcox
If
x 5, then x
If
x
5.
5, then x 5.
To solve
3 x 4 change to
1x 3 4
1x 7
x
7
1x 3 3 4 3
1x / ( 1) 7 / ( 1)
Free Pre-Algebra
Lesson 24  page 6
Lesson 24: Equations with Negatives
Worksheet
Name ________________________________________
Solve the equations. The answers may be positive or negative whole numbers or fractions.
8a 5
2b 7 19
13
14 d
21 3(c 9)
k 7
2
3m
5
1
4 3x 2x
18 2
x
2
45
7
2
16
Choose one of the problems above and check the answer in the original equation.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 24  page 7
Lesson 24: Equations with Negatives
Homework 24A
Name ______________________________
1. Translate the words to
mathmatical notation.
2. Find the absolute value:
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.
5
a.
3. Compare with > or <.
4. Compare with <, >, or =.
a. 7
a. 7
0
b. 0
b. 7
3
c. 9
c. 7
10
d.
d. 7
( 1)
10
1
b. (7 1)
1 7
(1 7)
c. (7)( 1) ( 1)( 7)
5. Simplify.
9 8
9( 8)
9 / ( 1)
( 1)( 2)( 3)
9 ( 8)
9 ( 8)
9 ( 1)
52
( 5)2
1 9
2 6 ( 5)
9 4 1 10
4
6. Simplify the algebraic expressions. Use super-efficient notation in your answers.
4x
8 5x
© 2010 Cheryl Wilcox
9 6x ( 5x )
9x 8y 7x 2(4 y )
Free Pre-Algebra
Lesson 24  page 8
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
16t
2
Convert –200ºC to Fahrenheit using the formula
F
192t
9C
5
32
The rocket reaches its greatest height after 6 seconds. What
is the height when t = 6?
Convert –193ºF to Celsius using the formula
C
What is the height when t = 12?
8. Evaluate each expression for the given values of the variable.
x x2
x 2 2x
9. Solve the equations.
4x
9
3
© 2010 Cheryl Wilcox
6 x
7
5(F
32)
9
Free Pre-Algebra
Lesson 24  page 9
Lesson 24: Equations with Negatives
Homework 24A Answers
1. Translate the words to
mathmatical notation.
2. Find the absolute value:
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
5
d.
4. Compare with <, >, or =.
a. 7 < 0
a. 7
0
b. 7 <
3
9
c. 7 >
10
b. 0
c. 9
4
c. When 5 positive particles are
combined with 4 negative particles
the result is that 1 positive particle
remains.
5
5
a.
3. Compare with > or <.
( 1)
1 7
6
b. (7 1)
c.
(7)( 1)
6
(1 7)
6
6
d. 7 < 10
1
1
( 1)( 7)
7 7
1
5. Simplify.
9 8
9( 8)
9
8
17
9 ( 8)
72
9 / ( 1)
9 8
9 ( 1)
9
(2)( 3)
6
1
1 9
8
4
4
9 8 17
( 5)2
52
8
( 1)( 2)( 3)
9 ( 8)
(5 • 5)
25
( 5)( 5) 25
9 4 1 10
2 6 ( 5)
8
4
26 5
2
2 • 11
(9 1) 10 4
(10 10) 4
2 11
22
4
6. Simplify the algebraic expressions. Use super-efficient notation in your answers.
4x
8 5x
4x 5x
1x 8
x 8
© 2010 Cheryl Wilcox
9x 8y 7x 2(4 y )
9 6x ( 5x )
8
9
9
9 x
6x
1x
5x
9x
2x
2x
7x
0y
8y
8y
Free Pre-Algebra
Lesson 24  page 10
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
16t
2
Convert –200ºC to Fahrenheit using the formula
192t
C
The rocket reaches its greatest height after 6 seconds. What
is the height when t = 6?
h
16(6)2 192(6)
16(36) 192(6)
576 1152
576 feet above ground
32
200
40
F
9( 200 )
32
5
360 32
328ºF
Convert –193ºF to Celsius using the formula
C
What is the height when t = 12?
h
9C
5
F
F
16(12)2 192(12)
16(144) 192(12)
2304 2304 0
The rocket hits ground.
5(F
32)
9
193
25
C
5(( 193) 32)
9
5( 225 )
9
125ºC
8. Evaluate each expression for the given values of the variable.
x x2
x 2 2x
9. Solve the equations.
4x
9
4x
6 x
3
9
4x
3
4x
6
x
6
4
© 2010 Cheryl Wilcox
9 9
4x / 4
3
2
6/4
3 9
7
x 6 7
4x 6
x
6
4
x 6 6 3 6
4x / 4 6 / 4
3
2
Free Pre-Algebra
Lesson 24  page 11
Lesson 24: Equations with Negatives
Homework 24B
Name ________________________________
1. Translate the words to
mathmatical notation.
a. Owing $75 is better than owing
$100.
2. Find the absolute value:
3. Compare with > or <.
a. 0
a. 9
7
b. 9
10
1
b.
b. The quotient of 4 and –4 is –1.
c. 9
c. 3
d. 0
c. The checking account had $30,
but there was a debit of $50,
resulting in an overdraft of $20.
5
4. Compare with <, >, or =.
a. 9 1
b. 9 1
4 2
4 2
d. 9
90
9 1
c.
4( 2)
4 ( 2)
( 2)( 3)( 4)
6 8
3
4 / ( 2)
( 4)2
42
3
32
6 2•3
( 3)2
6. Simplify the algebraic expressions. Use super-efficient notation in your answers.
3x
7x
4x
© 2010 Cheryl Wilcox
6 y 2 5y 3
1 ( 9)
9
5. Simplify.
4 2
9 ( 1)
3(x 2)
1 ( 9)
Free Pre-Algebra
Lesson 24  page 12
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
16t
2
Convert –300ºC to Fahrenheit using the formula
F
224t
9C
5
32
The rocket reaches its greatest height after 7 seconds. What
is the height when t = 7?
What is the height when t = 14?
Convert –301ºF to Celsius using the formula
C
8. Evaluate each expression for the given values of the variable.
3 3x
3x 2 3
9. Solve the equations.
8y 6 5
© 2010 Cheryl Wilcox
1
7 x
5(F
32)
9