Chapter 4

Equations and

Inequalities

CHAPTER 4

Look at this picture.The paper bag on the left contains a

certain number of orange blocks.Assuming that the paper bag weighs

nothing, can you determines how manyblocks are in the orange bag? What do you think we could do?

One way is to remove four orangeblocks from each side of the balance. The bag contains three blocks.

In algebra, an EQUATION is like a balance. The weights on each side of the scale are equal.

You can use algebra tiles to model and solve equations.

4-1-C: EXPLORE – SOLVE EQUATIONS USING ALGEBRA TILES

Solve using algebra tiles.1. Model the equation.2. Remove the same number of 1-tiles from each side of the

mat so that the variable is by itself on one side.3. The number of 1 tiles remaining on the right side of the

mat represents the value of x.

So, x = 3.


1x 1

1

1

1

1

11

1

1

1

A zero pair is a number paired with its opposite, like 2 and -2. You can add or subtract a zero pair from either side of an equation without changing its value, because the value of the zero pair is zero.

Solve with algebra tiles.1. Model the equation.2. Add two negative tiles to the left side

of the mat and two negative tiles to theright side of the math to form zero pairson the left.

3. Remove all of the zero pairs from the left side. There are 3 negative tiles on the right side of the mat.

So, x = -3.

Now, with your partner, work with your tiles to complete book page 207. #1-12. Be sure to write in COMPLETE SENTENCES when answering #9-12.


An EQUATION is a sentence stating two quantities are equal.

We can solve equations using INVERSE OPERATIONS. An inverse operation is an operation used to UNDO another

operation. What is the inverse operation of addition? What is the inverse operation of multiplication?

SUBTRACTION PROPERTY OF EQUALITY If you subtract the same number from each side of an

equation, the two sides remain equal. Symbols: If. Examples:

4-1-D: SOLVE ONE-STEP ADDITION AND SUBTRACTION EQUATIONS

Let’s look at three more examples before we try it on our own.

Now you try!


a = 7r = 7x = -3y = -4

If you are ever stuck and do

not know what to do, ask yourself these three questions:1. What is my variable?

2. What is being done to my variable?3. How can I UNDO what is

being done to my variable?

ADDITION PROPERTY OF EQUALITY If you add the same number to each side of an equation,

the two sides remain equal. Symbols: If. Examples:

Let’s look at a few more examples:


Now you try!

REAL WORLD Examples The highest recorded temperatures in Warsaw, Missouri, is

118°F. This is 158° greater than the lowest recorded temperature. Write and solve an equation to find the lowest recorded temperature.

The average span of a tiger is 17 years. This is 3 years less than the average lifespan of a lion. Write and solve an equation to find the average lifespan of a lion.


s = 17w = 18x = 7y = 11

On your own, practice solving addition

and subtraction equations by completing

#1 - 8 on page 211 in your math book.

The expression means 3 times the value of x. The numerical factor of a multiplication expression like 3x is called a coeffi cient. (In other words, the coeffi cient is the number attached to a variable.) So 3 is the coeffi cient of 3x.

Sometimes, you can use a drawing to represent an equation.

Look at This means “three groups of a something equal twelve. Start with three groups and then evenly divide up the

twelve. COUNT WITH ME! So x = 4.

Draw a picture for these:

4-2-B: SOLVE ONE STEP MULTIPLICATION AND DIVISION

EQUATIONS

DIVISION PROPERTY OF EQUALITY If you divide each side of an equation by the same nonzero

number, then two sides remain equal. Symbols: If. Examples:

Remember your three questions:1. What is my variable?2. What is being done to my variable?3. How can I UNDO what is being done to my variable?


EQUATIONS

c

Check out a few more examples:

Now try some on your own.


EQUATIONS

e = 3j = -9x = -5y = 4

MULTIPLICATION PROPERTY OF EQUALITY If you multiply each side of an equation by the same number,

then two sides remain equal. Symbols: If. Examples:

Check out a few examples before you try some on your own..


EQUATIONS

Now try to solve some on your own!

REAL WORLD EXAMPLES Mrs. Bybee’s car can travel an average of 24 miles on each

gallon of gasoline. Write and solve an equation to find how many gallons of gasoline she will need for a trip of 348 miles.

A Fitch ferret has a mass of about 2 kilograms. To find its weight in pounds p, you can use the formula . How many pounds does a Fitch ferret weigh? A FORMULA is an equation that shows the relationship among

certain quantities.


EQUATIONS

m = 70h = -36x =150y = 80

On your own, practice solving multiplicationand division equations by completing #1 - 8 on page 218 in your math book.

Mr. Hoff mann rides his bike for hour after school. Then he rides his bike for another hour before dinner. In all, he spends or 1 hour riding his bike. Copy and complete the table below.

What do you notice about the table?Two numbers with a product of 1 are called

MULTIPLICATIVE INVERSES, or RECIPROCALS. INVERSE PROPERTY OF MULTIPLICATION

The product of a number and its multiplicative inverse is 1.

4-2-D: SOLVE EQUATIONS WITHRATIONAL COEFFICIENTS

Find the multiplicative inverse of the following numbers:

8

Sometimes the coeffi cient of a term in a multiplication problem is a rational number. If the coeffi cient is a decimal, divide each side by the

coeffi cient.

If the coeffi cient is a fraction, multiply each side of the equation by the reciprocal of the coeffi cient.


Let’s practice with both decimal and fraction coeffi cients!

REAL WORLD EXAMPLES Suppose the ice cream cones cost $2.80 each. How many ice

cream cones could Miss Holloway buy with $42? Wilson has 9 pounds of trail mix. How many pound bags of

trail mix can he make?


x = 5t = d =7a = 2b = -6g = h = -3c =

On your own, practice solving equations with rational coefficients by completing#1 - 12 on page 224 in your math book.

A TWO-STEP EQUATION has two diff erent operations. To solve a two-step equation, UNDO the operations in

REVERSE order of the ORDER of OPERATIONS.

4-3-B: SOLVE TWO STEP EQUATIONS

In order of operations,

the multiplication

would be evaluated first. But

because we are doing

things in the reverse order,

we must UNDO the

addition first.

After the first step, we’re left with a one-step division

equation!

Here are a few more examples before you try some on your own.

Key ConceptStep 1: Undo the addition or subtraction first.

Step 2: Then undo the multiplication or division.


STEP 1

STEP 2

Now you try!

REAL WORLD EXAMPLE A fitness club is having a special offer where you pay $22 to

join plus a $16 monthly fee. You have $150 to spend. Write and solve an equation to find how many months you can use the fitness club.


x = 3y = - 3m = - 4r = 2.4s = - 4q = 48d = 8p = - 9

On your own, practice solving two-step

equations by completing

#1 - 7 on page 232 in your math book.

As you discovered in the Geology Rocks Equations activity, you can use the properties of equality with variables on each side. The new equation is equivalent to the original equation.

Remember the four properties of equality:1. Subtraction Property of Equality2. Addition Property of Equality3. Division Property of Equality4. Multiplication Property of Equality

Key Concepts: We have to use the equality properties in order to get the

variable on to ONE side of the equation. We can only solve the equation if we can ISOLATE the variable.

Remember that what you do to one side of the equation, you MUST do the EXACT SAME THING to the other side of the equation.

4-3-D: SOLVE EQUATIONS WITH VARIABLES ON BOTH SIDES

Let’s look at some examples

TIP: Work with the variables fi rst. Once you

can get those to one side, then you’re left with two-step equations.

SIGNS: Sometimes, equations will be written diff erently. A good strategy is to always look at the sign in front of numbers.


It does not matter which side your variable ends up on. When making your fi rst move, ask yourself, which variable would be easier to move. For example, look at the equation .

Would it be easier to ADD 4x to both sides? Or would it be easier to SUBRACT 2x from both sides?

In the end, it does NOT matter which you do, you will end up with the same answer either way.

Now you try!


x = 6x = 24x = 4.2x = - 10x = ¼x = -6

REAL WORLD EXAMPLES Bill averages 10 points a game and has 110 points for the season.

Aaron averages 12 points a game and has 96 points for the season. Write and solve an equation to find how many games it will take until they tie in points scored if they continue at the same rate. Remember to ask yourself:

What stays constant no matter what? What is dependent upon the number of games?

We know that Bill already has 110 points for the season – so that’s constant. He averages 10 points PER game, so we know we’ll multiply 10 by the number of games, g.

We know that Aaron already has 96 points – so that’s constant. He averages 12 points a game, so we know we’ll multiply 12 by the number of games, g.

Since we want to find the SAME g for both players, we can set these equal to each other and solve.

On your own, practice solving equations by completing numbers 1-10 on page 238 in your math book.


An INEQUALITY is a mathematical sentence that compares two quantities. - LESS THAN, and LESS THAN OR EQUAL TO

(Looks like a crooked L) - GREATER THAN, and GREATER THAN OR EQUAL TO

When we are solving inequalities, we are finding values that make the inequality true. We want to identify ALL the numbers that make then

inequality true.

KEY CONCEPT When you add or subtract the same number from each side of

an inequality, the inequality remains true. Sound familiar? Inequalities are solved using the same properties

of equality that we use for equation.

4-4-B: SOLVE INEQUALITIES BY ADDITION OR SUBTRACTION

InequalitiesWords • is less than

• is fewer than• is greater

than• is more than• exceeds

• is less than or equal to

• is no more than

• is at most

• is greater than or equal to

• is no less than

• is at least

SymbolsCheck out a few examples!


Once we solve inequalities, we also have to graph the solution on a number line. OPEN AND CLOSED CIRCLES

When graphing inequalities, an open circle is used when the value should not be included in the solution, as with and inequalities.

A closed circle indicates the value is included in the solution, as with and inequalities.

Let’s practice graphing our answers from the previous examples. , or

Notice that either way we write the inequality, the n is still greater.


Now you try! Solve each inequality. Graph the solution set on a number line.

Write an inequality and solve each problem Four more than a number is more than 13. The sum of a number and 19 is at least 8. Eight less than a number is less than 10. The difference between a number and 21 is no more than

14.


or

or

or

Similar to equations, we will also use the Division and Multiplication Properties of Equality to solve inequalities. What you do to one side, you must to do the other…

Key Concept When you multiply or divide each side of an inequality by a

positive number, the inequality remains true.

When you multiply or divide each side of an inequality by a negative number, the inequality sign must be reversed (or flipped) for the inequality to remain true.

4-4-C: SOLVE INEQUALITIES BYMULTIPLICATION AND DIVISION

Check out a few more examples:

Now try some on your own!

4-4-C: SOLVE INEQUALITIES BYMULTIPLICATION AND DIVISION

or

FLIPPED

FLIPPED

or

Two-step inequalities are solve similarly to 2-step equations. Remember:

Undo operations in the reverse order of Order of Operations Remember to FLIP the inequality sign if you multiply or divide by a

negative number.

Solve each inequality. Graph the solution on a number line.

SOLVING 2-STEP INEQUALITIES

v > 5r > 9x ≤ 8m ≥ -6n > 5x ≥ 15x > -9m < -4

Chapter 4

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Transcript of Chapter 4