Grade 9 Algebra by Elaine Ernst Schneider

Algebra provides the basics for all higher math. You will work with numbers and letters (variables) to form sentences (expressions) that you can solve. The best way to learn math is by practicing it, so each lesson will include exercises using the skills learned.

A place to begin:

Letters in math are called
variables. They can stand for different numbers at different times.

A mathematical sentence
is called an expression. It can include numbers, variables, signs of operation,
and symbols of inclusion.

Signs of operation tell
you what to do to the sentence. The four operations are addition, subtraction,
multiplication, and division.

Symbols of inclusion are
parentheses ( ) and brackets [ ].

An important caution:

Be very neat in your calculations. Many an algebra problem is missed because the student misread what he or she had written or did not "line up" the column correctly for subtraction or division. Always double check operations. You don't want to miss a problem because you added incorrectly.

Let's Get Started:

To "evaluate" an expression means to find its value, or to solve it. The first rule to learn about algebra is "what to do when." The order in which an expression's operations are done can completely change the answer.

When evaluating an algebraic expression, first look for the symbols which show the innermost work. That can be expressed by use of parentheses or brackets. If BOTH parentheses and brackets are present, the parentheses are usually the innermost and should be worked first.

Here is an example:

24 + [46 - (2 X 11)]

24 + [46 - 22]

24 + 24

48

Now it's time for you to try a few.

EXERCISE:

9 - (4 X 2)

(9 - 4) X 2

(9 - 4) X (2 X 1)

48 - [42 - (3 X 9)]

63 - [8/2 + (14 - 10)]

(Note: 8/2 is the same as
8 divided by 2, just like in fractions.)

[800/ (200 X 4)]

28 + [ 10 - (4 + 2) ]

(11-5) X (10 + 14)

125 / ( 5 X 5) (Remember
from number 5? / = divided by.)

[28 - (4 X 5)] - 4

ANSWER KEY

Question# | Answer |

1...
2... 3... 4... 5... 6... 7... 8... 9... 10... |
1
10 10 33 55 1 32 144 5 4 |

In the first lesson, you learned that numbers and variables form sentences, or algebraic "expressions." When you take information from a sentence and turn it into a mathematical expression, it is called "translating."

A place to begin:

- When you write algebraic expressions, use +, -, and = signs. For division, use / , the same way you know that when you see a fraction, it means to divide the top number by the bottom number.
- For multiplication, write the expression with no symbol or sign between them as the X (multiplication) symbol can be confused with the variable x. For example 3 times the variable y should be written 3y. You can also use parentheses to indicate multiplication. This is especially useful in longer problems such as (3y)(4-2x).
- When you want to multiply something AFTER another expression has been done first, use parenthesis. For example, if you want to add x and y and THEN multiply the result by 7, write it this way: 7(x + y).
- To translate from language to a math expression, read the sentence carefully. Then decide what operations it will take to reach a solution. Write this into an algebraic expression.

Here is the word problem:

I am stocking shelves at a T-shirt store. I have been given eight boxes of T-shirts. There are 25 T-shirts in each box. My instructions are to take one T-shirt from every box and set it aside to be given to the poor. How many T-shirts will be left for me to put on the store shelves after I sort out the T-shirts to be given to the poor?

Here is the algebraic expression:

8(25-1)

8(24)

192

If you don't know the number of T-shirts in each box, then substitute the variable x for that. Then the expression would look like this:

8(x - 1)

Now
it's time for you to try a few.
**Assignment(s) including
Answer key:**

EXERCISE:

- Katie is 21 years old. Write an algebraic expression that tells how old she will be in five years. Let x represent that age. Solve from your expression.
- One hamburger costs x cents. Write an algebraic expression that tells how much 4 hamburgers will cost.
- Sandy is 21 years old. Al is five years older. How old will Al be in five years? Let x represent Al's age in five years. Solve for x.
- I have ten hats that cost $47.50 for all of them. Write an algebraic expression to show how much one of those hats cost, letting y represent the answer. Solve for y.
- I have ten hats that cost $47.50 for all of them. How much will I have to sell EACH hat for to make a profit of $1 on each? Set it up in an algebraic expression to solve.

Question | Answer |

1.. | 21 + 5 = x
26 = x |

2.. | 4x |

. | |

3.. | (21 + 5 = Al's
age now)
(21 + 5) + 5 = x (Al's age in 5 years) |

. | |

4.. | $47.50 / 10 = y
$4.75 = y |

. | |

5.. | (47.50 / 10 ) + $
1 = x
$4.75 + $ 1 = x $5.75 = x |

A quick review:

1. When you write algebraic expressions, use +, -, and = signs. For division, use / , the same way you know that when you see a fraction, it means to divide the top number by the bottom number.

2. For multiplication, write
the expression with no symbol or sign between them as the X

(multiplication) symbol
can be confused with the variable x. For example 3 times the variable y
should be written 3y. You can also use parentheses to indicate multiplication.
This is especially useful in longer problems such as (3y)(4-2x).

3. When you want to multiply
something AFTER another expression has been done first, use

parenthesis. For example,
if you want to add x and y and THEN multiply the result by 7, write it
this way: 7(x + y).

4. To translate from language to a math expression, read the sentence carefully. Then decide what operations it will take to reach a solution. Write this into an algebraic expression.

Something new:

1. To take something OUT of parenthesis, do the operation one number at a time. For example, 7(x+y). First, multiply 7 times x. Then multiply 7 times y. The result is 7x + 7y.

2. When you solve for x, you want to "isolate" the x on one side of the equal sign. To do this, use the opposite sign of the number you want to move and do the same thing to BOTH sides of the equation.

For example: 8x + 2 = 50

8x + 2 - 2 = 50 - 2 (Subtract 2 from both sides of the = sign)

8x = 48 (Divide by 8 to solve because that is opposite of multiplication)

x = 6

And now a fun problem to
make you really think. Solve using algebra. Translate into an expression
and solve.

You can do it!

**Assignment(s) including
Answer key:**

George is 4 years older than Jon, who is 4 years older than Jim, who is 4 years older than Sam, who is 1/2 the age of George. How old is each boy? Hint: let x represent George's age.

Answer Key

If x is George's age, then
x - 4 is Jon's age because George is 4 years older than Jon. George is
8 years older than Jim, making Jim's age x - 8. George is 12 years older
than Sam, so Sam's age can be represented as

x - 12.

We also know that Sam is 1/2 the age of George.

So, set up the expression:

(x - 12) = 1/2 x

To solve for x, you first
need to put 1/2x into a different form.

Multiply 1/2 by x.

x is the same as x over 1 if you put it into fraction form, or x/1. So, when you multiply 1/2 by x, it would actually look like (1/2) (x/1). When you multiply fractions, you multiply the numerators (1)(x) and then the denominators (2)(1). The result is x/2. Now rewrite the expression using x/2 instead of 1/2x.

(x-12) = x/2

We learned that when you solve an expression, you want to "isolate" the variable. To do this, multiply BOTH sides by 2.

(x-12) = x/2

2(x-12) = 2(x/2)

2x -24 = x

Now, you need to get both variables on the same side of the equal sign, so subtract 2x from BOTH sides:

2x - 2x - 24 = x - 2x

-24 = -1x (Divide -24 by -1 to solve for x)

24 = x

therefore, George is
24. Jon's age is represented by x - 4 or 20.

Jim is x - 8, or 16. And
Sam is x - 12, or 12.

You can further check the
problem by multiplying Sam's age by 2, to get George's age, because the
problem told us that Sam is half the age of George.

In the first lesson, you
learned that numbers and variables form sentences, or algebraic "expressions."
When you take information from a sentence and turn it into a mathematical
expression, it is called "translating." In another lesson, you learned
that when you write algebraic expressions, use +, -, and = signs; and for
division, use / , the same way you know that when you see a fraction, it
means to divide the top number by the bottom

number. Then, for multiplication,
we learned to write the expression with no symbol or sign between them
(such as 3a), with an X , or using parentheses. The parenthesis is especially
useful in longer problems such as (3y)(4-2x).

But what if there are no parentheses or brackets? How do you know which to do first? Add, divide, multiply? Which one?

When there are no other indications as to which computation to do first, mathematicians follow The Order of Operations rule. Multiply, Divide, Add, Subtract.

A good way to remember the Order of Operations is to think about My Dear Aunt Sally. The M is for multiply, the D in Dear is for Divide, the A is for Add, and the S in Sally is for subtract.

HINT: If the problem has only multiplication and division, then work from left to right.

HINT:
If the problem has only subtraction and addition, then work from left to
right.

Let's Get Started:

13 - 2 X 5

If I just do the math from left to right, I would say 13 - 2 = 11. Then 11 times 5 = 55.

But the Order of Operations tells me to multiply first! So, 2 X 5 = 10. Then, I subtract 10 from 13 and get 3.

You can see that which order you choose makes a BIG difference in the answer! That's why it's important to follow the Order of Operations.

Now it's time for you to
try a few.
**Assignment(s) including
Answer key:**

1. 10 - 4 + 3

2. 10 + 4 X 2

3. (5 X 4) -15 + 2

4. 12 - 2(3 + 1)

5. 18 + 2(3)

6. (12 - 2)(3 + 4)

7. 4 X 3 + 5

8. 24 - 6 +2

9. 24 - 6 X 2

10. 36/9 - 2

**ANSWER
KEY**

1. 9

2. 18

3. 7

4. 4

5. 24

6. 70

7. 17

8. 20

9. 12

10. 2

Review:

In the first lesson, you
learned that numbers and variables form sentences, or algebraic "expressions."
When you take information from a sentence and turn it into a mathematical
expression, it is called "translating." In another lesson, you learned
that when you write algebraic expressions, use +, -, and = signs; and for
division, use / , the same way you know that when you see a fraction, it
means to divide the top number by the bottom

number. Then, for multiplication,
we learned to write the expression with no symbol or sign between them
(such as 3a), with an X , or using parentheses. The parenthesis is especially
useful in longer problems such as (3y)(4-2x).

Then you learned how to work problems where there are no parentheses or brackets, using the Order of Operations rule. Multiply, Divide, Add, Subtract.

Now let's put all of this
to use and Yes, that's right! We're going to take English sentences - WORDS
- and turn them into algebraic expressions.

Let's Get Started:

Subtract seven from twenty-one, then add three.

The algebraic expression is: 21 - 7 + 3

No parentheses are needed
because the Order of Operations tells us that addition and subtraction
are done in order from left to right.

Now it's time for you to
try a few. Remember your terms: subtract, sum, product, division, multiply,
quotient. You may have to use parenthesis on some of them.

**Assignment(s) including
Answer key:**

1. Subtract 2 from x; then add y.

2. Subtract the sum of 2 and y from x.

3. Divide 10 by 3; then multiply by 5.

4. Divide x by the product of 3 and z.

5. Multiply x by 3; then add y.

6. Add x and 3; then multiply by y.

7. Subtract the product of 5 and x from 7.

8. 5 more than the product of 3 and c.

9. 13 less than the quotient 5 divided by p.

10. 4 times the sum of 10 and x.

**Answer
Key**

1. x - 2 + y

2. x - (2 + y)

3. 10/3 X 5

4. x/32

5. 3x + y

6. (x + 3) y

7. 7 - 5x

8. 3c + 5

9. 5/p -13

10. 4(10 + x)

In the last lesson, you learned
to write expressions and to find the values of expressions.

When two expressions can
be written to balance or equal one another, it is called an

equation. In other words,
whatever is expressed on one side of the equal sign is calculated to be
the exact same value as whatever is on the other side of the equal sign.
IN other words, the left member and the right member of the equation "balance."

Examples of equations:

5 + 7 - 2 = 2(5) (the answer on the left side is 10 and the answer on the right side is 10)

100/25 = 347 - 343 (the answer is 4 on the left of the equal sign and 4 on the right also)

Equations involving unknown variables are solved by balancing the left member and the right member. For example, in the equation y + 5 = 12, I know that the left member must equal 12 for it to balance with the right member. This means y would be 7 because 7 + 5 = 12. That one is easy enough to do the math without a formal procedure. But because algebraic equations can become much longer and more calculated, a system is necessary to solve for variable and balance equations.

Let's take the above equation:

y + 5 = 12

The proper procedure in solving for y is to "isolate" y. This means we want y to stand by itself on one side of the equal sign.

Now, keep in mind that we want every thing to "balance" on either side of the equal sign. This means whatever I do to the left member, I must do to the right member. So, to isolate y and solve the equation, I must "move" the 5 to the other side of the equal sign. To do this, I must make it zero on the left side of the equation by subtracting 5:

y + 5 = 12

y + 5 - 5 = 12 - 5

y + 0 = 12 - 5

y = 7

Try solving for the variable
by keeping each side equal. Remember, the idea is to "balance"

the equation, so what you
do to one side, you must do to the other.

100/5 = y + 2

100/5 -2 = y +2 - 2

20 - 2 = y

18 = y

Now it's time for you to try a few...

**Assignment(s) including
Answer key:**

EXERCISE:

1. y - 10 = 17

2. c + 4 = 29

3. 5y = 90

4. 1/3 t = 29

5. 6y = 72

6. 1/9 g = 58

7. 4y = 100

8. 12a = 132

9. r + 9 = 48

10. x + 79 = 422

ANSWER KEY:

1. 27

2. 25

3. 18

4. 87

5. 12

6. 522

7. 25

8. 11

9. 39

10. 343

In the last lesson, you learned to balance an equation, making what is on one side of the equal sign equal to what is on the other side. In other words, the left member and the right member of the equation "balance."

Examples of equations:

5 + 7 - 2 = 2(5) (the answer on the left side is 10 and the answer on the right side is 10)

100/25 = 347 - 343 (the answer is 4 on the left of the equal sign and 4 on the right also)

You also learned to isolate variables. Equations involving unknown variables are solved by balancing the left member and the right member. In the equation, y + 5 = 12, the proper procedure in solving for y is to "isolate" y. This means we want y to stand by itself on one side of the equal sign.

Now, keep in mind that we want every thing to "balance" on either side of the equal sign. This means whatever I do to the left member, I must do to the right member. So, to isolate y and solve the equation, I must "move" the 5 to the other side of the equal sign. To do this, I must make it zero on the left side of the equation by subtracting 5:

y + 5 = 12

y + 5 - 5 = 12 - 5

y + 0 = 12 - 5

y = 7

Equations can also be used to solve puzzles. You must set up an equation that "matches" in numbers what the puzzle expresses in words. Then, solve for the variable. That will be the answer to the puzzle.

Why don't you try a few?

**Assignment(s) including
Answer key:**

1. Farmer Brown told Bob
and Sue that they could pick apples from his tree, but that neither of
them could take more than 20. They worked for a while, and then Bob asked
Sue, "Have you picked your limit yet?"

Sue replied, "Not yet. But
if I had twice as many as I have now, plus half as many as I have now,
I would have my limit." How many did Sue have?

2. A little boy was told not to eat the grapes from the vine for fear that he would eat too many and get a stomachache. Sneaking out to the grape arbor when his mother wasn't looking, the little boy ate grapes for five days, each day eating 6 more than the day before. In fact, after five days, the little boy was so sick that he had to confess to his mother that he had eaten 100 grapes. How many grapes did the little boy eat on EACH of the five days?

3. How high is a tree that
is 15 feet shorter than a pole three times as tall as the tree?

ANSWER KEY:

1. Let x = the number of apples she had.

2x + 1/2 x = 20

(2x as a fraction with 2 as the denominator would be written 4x/2.)

4x/2 + 1/2 x = 20

5x/2 = 20

5x/2 X 2 = 20 X 2

5x = 40

5x/5 = 40/5

x = 8 apples

2. Let x = number of grapes the little boy ate the first day

x + 6= number of grapes eaten the second day

x + 6 + 6 = number of grapes eaten the third day

x + 6 + 6 + 6 = number of grapes eaten the fourth day

x + 6 + 6 + 6 + 6 = number of grapes eaten the fifth day

Five days' worth of grapes = 100 in all. Therefore, the equation to set up is:

x + (x + 6) + (x + 6 + 6) + (x + 6 + 6 + 6) + (x + 6 + 6 + 6 + 6) = 100

x + x + 6 + x + 12 + x + 18 + x + 24 = 100

5x + 60 = 100

5x + 60 - 60 = 100 - 60

5x = 40

5x/5 = 40/5

x = 8 grapes eaten on the first day

(Now put 8 in place of x in all the other expressions.)

x + 6 = 14 grapes eaten the second day

x + 6 + 6 = 20 grapes eaten the third day

x + 6 + 6 + 6 = 26 grapes eaten the fourth day

x + 6 + 6 + 6 + 6 = 32 grapes eaten the fifth day

(To check, add 8, 14, 20, 26, ad 32. They equal 100.)

3. Let y = number of feet in height of tree

3y would equal the number of feet in height of the pole

Set up the equation:

y = 3y - 15

y - 3y = 3y - 3y - 15

-2y = -15

-2y/-2 = -15/-2

y = 7.5 feet

3y = 22.5 feet

In the last lesson, you learned that equations can be used to solve puzzles. For years, mathematicians have been solving the following puzzle to the amazement of their students. Not only is it a fun "trick," but it is based on a sound mathematical principle that makes it work:

By using your house number, a mathematician can calculate your age!

Here's how it works:

1.Double your house number.

2.Add 5.

3.Multiply by 50.

4.Add your age.

5.Add the number of days in a year (365).

6.Take that number and subtract 615.

7.Pretend it is an amount of money and position the decimal point for dollars
and cents.

Answer: the dollars will
be your house number and the cents will be your age.

But now the big question is: HOW does it work? Give it a little thought before looking at the answer.

ANSWER:

To understand why the puzzle works, you must solve it algebraically.

Let x = your house number.

Let y = your age.

Let A = answer

Using the numbered list of instructions, set up the problem as an algebraic equation. It would look like this:

A = 50(2x + 5) + y + 365 - 615

100

A = 100x + 250 + y +365 -615

100

A = 100x + y

100

A = x + y/100

Therefore,

x = the house number (You started with this number.)

y/100 = your age (When you divide by 100, you move the decimal point two
places to the left.)

Different kinds of numbers are used to stand for different things. There are positive numbers and there are negative numbers. Positive numbers are more than zero and negative numbers are less than zero.

The
best way to "see" negative and positive numbers is to look at a number
line:

__/_____/_____/_____/_____/_____/_____/_____/_____/_____/_____/_____/_____/_____/_____/__

-7
... -6 ..... -5 .... -4 ...... -3 ...... -2 ..... -1 ..... 0 ...... 1 ......
2 ...... 3 ...... 4 ...... 5 ...... 6 ...... 7

When you use a number line, you add positives as you go to the right and you add negatives as you move to the left. For example, 3 + -2 means that you start at 3 and move two places to the left. The answer is 1. To add a positive to a negative, you work from left to right. For example, -4 + 3 is -1. You get that by starting at -4 and moving three places to the right. To add two negatives, move to the left. For example, -1 + -3 = -4. You arrive at that answer by starting at -1 and counting three places to the left.

Now you try:

1
. 1 + -1

2.
4 + -2

3.
2 + 4

4.
-2 + -5

5.
-5 + 4

6.
- 3 + 5

7.
-2 + -2 + -1

8.
-5 + 6

9.
-7 +2

10
-4 + -2 + 2

Now try a few that are beyond the scope of the number line shown above. You can do it! Just use the principles you have learned.

- -15 + 13
- -39 + -40
- -23 + 20
- 2 + -11
- -13 + -4

ANSWERS:

1.
0

2.
2

3.
6

4.
-7

5.
-1

6.
2

7.
-5

8.
1

9.
-5

10
-4

********

- -2
- -79
- -3
- -9

-17

Algebra Lesson #10 - Absolute Values

In lesson 9, we learned about negative and positive numbers by using a number line. Negatives go left from the zero, and positives are counted off to the right. To add a negative and a positive, you start at one of the numbers and then count either to the left or the right depending on the sign. For example, to add -2 and 3, you can start at -2 and count to the right three places. You would end up at 1. -2 + 3 = 1

There
is a term that is used in math to represent the distance that a number
is from zero. That term is *absolute value. *The absolute value of
3 is three. The absolute value of -6 is six. This means that in terms of
absolute value, -6 is larger than 3. In other words, 6 takes up more places
on the number line. Look at the number line below. The red numbers represent
the numbers of places it takes to reach -6 from zero. The blue color represents
three. You can see that the red line is much longer than the blue one.

/_____/ _____/_____/_____/_____/_____/_____/_____/_____/_____/_____ /_____/

-6
... -5 …... -4 ….. -3 ….. -2 ….. -1 ….. 0 .…. 1
...… 2 ...… 3 …....4 …... 5 ..…. 6

The
concept of absolute value is important when adding positive and negative
numbers. For example, if you have -7 + 4, you may wonder which number to
use to begin the addition process. The number four is positive, so you
might think that it represents "more." Anyone would rather have $4 in their
checking account that be in the hole $-7; right?! But in order to do the
math, we must concern ourselves only with *absolute value. *The absolute
value of seven is greater than the absolute value of four. Therefore, we
begin with -7 and treat it as the larger number.

Here's the problem:

-7 + 4 =

Subtract the values. 7 - 4 = 3. To determine which sign to use, you must consider absolute value. Since 7 is larger in absolute value, its sign will dominate. The answer will be -3.

-7
+ 4 = -3

Suppose you have two negative numbers:

-7 + -2

Add the values. 7 + 2 = 9. Since you added absolute values, the answer is - 9.

-7 + -2 = -9

Now you try:

- -7 + -9
- -7 + 9
- -6 + 5
- -2 + -5
- -5 + 4
- - 3 + -5
- -5 + 2
- -15 + 6
- -17 +22
- -4 + -21

- -16
- 2
- -1
- -7
- -1
- -8
- -3
- -9
- 5

-25

Algebra Lesson 11 - Additive Inverses

Additive Inverses are

Another way to write the opposite of –8 is to write –(-8). To subtract a negative 8 is the same as making it a positive 8, or finding its opposite.

You can do the same thing with variables. For example, -(-x) = x.

Using this principle, let's
add a negative number to a positive one: 8 + (-3) = 8 – 3 = 5. This could
be read as 8 plus a negative 3 OR 8 minus 3 OR 8 plus the *opposite*
of 3.

In algebra, subtracting a
number can also be described as *adding its opposite.*

For example, x – y = x + (-y).

OR

8 – (-5) = 8 + 5 = 13

Now, let's turn things around a bit. Try this one:

14 – 28 = x

How can you rewrite that to use what you've learned about opposites?

14 + (-28) = x

–14 = x

Let's try one more:

11 – (-3) – 4 = x

11 + 3 – 4 = x

14 – 4 = x

10 = x

1. 5 – 7

2. 8 – (-10)

3. – 3 – 7

4. – 7 – 9

5. 7 + (-3)

6. 25 – 250

7. 5.8 – 2.3

8. 2.3 – 5.8

9. (34 – 13) – (15 – 17)

10. 11 – 5 – [6 + (-13)]

Answer key:

1. –2

2. 18

3. –10

4. –16

5. 4

6. –225

7. 3.5

8. –3.5

9. 23

10 .13

Algebra Lesson 12: Multiplying Positives and Negatives

As we learned last lesson: Additive Inverses are opposites. Two numbers are opposites if their sum equals zero. For example, -8 and 8 are additive inverses because their sums total zero. This makes them opposites. You can do the same thing with variables. For example, -(-x) = x.

When this principle is used in multiplication, these rules of thumb emerge:

1. A negative times a negative gives a positive answer.

2. A negative times a positive is negative.

3. A positive times a positive is a positive result.

Examples

(-6)(8) = -48

(-3)(-12) = 36

(12)(11) = 132

Now, what happens when there
are three numbers to multiply? Simply work in sequence, following the rules
you have learned.

Example: (-5) (-4) (-20)

(20) (-20) [multiply –5 times –4 to get positive 20]

-400 [multiply positive 20 from last step by –20 to get –400]

And what about exponents?
Just write them out and follow the rules.

Example: -45 = (-4) (-4) (-4) (-4) (-4)

(16) (-4) (-4) (-4)

-64 (-4) (-4)

256 (-4)

-1024

Exercises:

1. (10) (-8)

2. (-5) (-26)

3. -53

4. (-2) (-3)2

5. (-1.5) (4)

6. (-5) (22) (-2)

7. (-3) (-5) (-4)

8. (-1) (-3)4

9. (-1)15

10. (-2)2 (-3)4

Answer key:

1. –80

2. 130

3. –125

4. –18

5. –6

6. 220

7. –60

8. –81

9. –1 *

10. 324

**Note:
Did you learn something about exponents? If the exponent is even, the answer
is positive. If the exponent is odd, the answer is negative.

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