Module 3 notes.pdf

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Chapter 6: Equations and Inequalities

Prior Knowledge Needed

1. Solve equations using Guess, Check and Revise

2. Solve equations mentally

3. Solve an equation using inverse operations Inverse operations undo other operations.

 To solve an addition equation you would use subtraction

 You can solve a subtraction equation by adding

 Use division to solve a multiplication problem

 Solve a division problem with multiplication

Solve 14 – p = 6

Guess the value of p, then check it. Try 7.

14 – p = 6 14 – 7 ≠ 6 revise

Try 6. 14 – p = 6 14 – 6 ≠ 6 revise

Try 8. 14 – p = 6 14 – 8 = 6 yes

The solution is 8 because replacing p with 8 results in a true sentence.

Solve 15 ÷ m = 5

Think 15 divided by what number is 5? 15 ÷ m = 5

15 ÷ 3 = 5 5 = 5

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6-1: Solve One-Step Addition and Subtraction Equations

Equation- A sentence stating that two quantities are equal. An equation must contain an equal

sign.

Solution- The value of a variable that makes an equation true.

Equivalent Equations – Equations that have the same solution.

To solve an addition equation, use the subtraction property of equality

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When solving for a variable, ask yourself these questions: 1. What variable am I solving for?

2. What side is that variable on?

3. What numbers are on that side that I need to get rid of to isolate the variable (get it alone)?

4. How can I get rid of those numbers using inverse operations? 5. Simplify an expression so there are no grouping symbols- 6. Undo addition or subtraction first

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6-2: Solve One-Step Multiplication and Division Equations

To solve an multiplication equation, use the division property of equality

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6-2: Solve Equations with Rational Coefficients

Solving Equations with rational numbers is similar to solving equations with whole numbers. The only difference is you now have to remember your rules for computing with integers,

fractions and decimals as well as how to solve the equation.

Trick: When you are solving a multiplication problem with a fractional coefficient there are 2 ways to solve the problem.

1. You can divide both sides by the fractional coefficient or

2. You can multiply both sides by the reciprocal of the fractional coefficient because it is the multiplicative inverse.

Ex. 1

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6-4: Solve Two-Step Equations

When solving for a variable, ask yourself these questions: Example

1. What variable am I solving for? x

2. What side is that variable on? or What side is the variable with the largest coefficient on? left 3. What numbers are on

that side that I need to get rid of to isolate the variable (get it alone)? -2 and 4

4. How can I get rid of those numbers using inverse operations? Add 2 and divide by 4

5. Simplify an expression so there are no grouping symbols- there are none 6. Undo addition or

subtraction first. I must add 2 to both sides 7. Undo multiplication and

division next. I must divide both sides into 4 equal groups.

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Example 1. What variable am

I solving for? y

2. What side is that variable on? the left

3. What numbers are on that side that I need to get rid of to isolate the variable (get it alone)? -7 & -2

4. How can I get rid of those numbers using inverse operations? Add 7, divide by -2

5. Simplify an expression so there are no grouping symbols. Lucky for me, there are no grouping symbols.

6. Undo addition or subtraction first. I must add 7 to both sides first.

7. Undo multiplication and division next. I must divide both sides by negative 2.

Example 1. What variable am

I solving for? r

3. What numbers are on that side that I need to get rid of to isolate the variable (get it alone)? 4 & 1/5

4. How can I get rid of those numbers using inverse operations? Subtract 4, divide by 1/5 or multiply by 5.

5. Simplify an expression so there are no grouping symbols. Lucky for me, there are no grouping symbols.

6. Undo addition or subtraction first. I must subtract 4 from both sides first.

7. Undo multiplication and division next. I must divide both sides by 1/5 or multiply both sides by 5

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6-5: More Two-Step Equations (with grouping symbols)

When you have grouping symbols, you can simplify the expression using the distributive property. (This will always get rid of parentheses)

Remember that an expression like 2(x + 6) = 12 means 2 times the sum of x and 6. Therefore 2 is a factor and the other factor is (x+6). To separate factors we divide. If the numerical factor is a factor of the the constant on the other side, you can just divide both sides by the numerical factor instead of simplifying using the

distributive property.

Example

1. What variable am I solving for? x

3. What numbers are on that side that I need to get rid of to isolate the variable (get it alone)? 2 & 6

4. How can I get rid of those

numbers using inverse operations?

Subtract 6, divide by 2.

5. There are grouping symbols, what should I do? 2 is a factor of 16 so I will just leave the grouping symbols for now.

6. Undo addition or subtraction first. I can’t because it is inside of parentheses.

7. Undo multiplication and

division next. I must divide both sides by 2.

8. Now I can undo the addition and subtraction. I must subtract 6 from both sides.

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Example 1. What variable am I solving for? c

2. What side is that variable on? the left

3. What numbers are on that side that I need to get rid of to isolate the variable (get it alone)?

.2 & 3

4. How can I get rid of those numbers using inverse operations? Add 3, divide by .2.

5. There are grouping symbols, what should I do? -10 can be divided by .2, so I will just leave the grouping symbols for now.

6. Undo addition or subtraction first. I can’t because it is inside of parentheses.

7. Undo multiplication and division next. I must divide both sides by .2.

8. Now I can undo the addition and subtraction. I must add 3 to both sides.

Example 1. What variable am I solving for? c

3. What numbers are on that side that I need to get rid of to isolate the variable (get it alone)? .2 & 3

4. How can I get rid of those numbers using inverse operations? Add 3, divide by .2.

5. There are grouping symbols, what should I do? –I can use the distributive property to simplify the expression and eliminate the parentheses

6. Undo addition or subtraction first. I must add .6 to both sides now (instead of 3).

7. Undo multiplication and division next. I must divide both sides by .2.

.2 (c – 3) = -10

.2c - .6 = -10

+ .6 +.6

.2c = -9.4

.2 .2

c = -47

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Example 1. What variable am I solving for? n

3. What numbers are on that side that I need to get rid of to isolate the variable (get it alone)? 2/3 & 6

4. How can I get rid of those numbers using inverse operations? Subtract 6, divide by 2/3 or multiply by 3/2.

5. There are grouping symbols, what should I do? 10 can be divided by 2/3, so I will just leave the

grouping symbols for now.

6. Undo addition or subtraction first. I can’t because it is inside of parentheses.

7. Undo multiplication and division next. I must divide both sides by 2/3 or multiply both sides by 3/2.

8. Now I can undo the addition and subtraction. I must subtract 6 from both sides.

Example 1. What variable am I solving for? n

3. What numbers are on that side that I need to get rid of to isolate the variable (get it alone)? 2/3 & 6

4. How can I get rid of those numbers using inverse operations? Subtract 6, divide by 2/3 or multiply by 3/2.

5. There are grouping symbols, what should I do? I can use the distributive property to simplify the expression and eliminate the parentheses.

6. Undo addition or subtraction first. I must subtract 4 from both sides now (instead of 6).

7. Undo multiplication and division next. I must divide both sides by 2/3 or multiply

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6-6: Solve Inequalities by Addition or Subtraction

Inequality- A mathematical sentence that compares quantities.

Graph with

Empty circle

Filled circle

Filled circle Addition Property of Inequality & Subtraction Property of Inequality

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6-7: Solve Inequalities by Multiplication or Division

Multiplication Property of Inequality

&

Division Property of Inequality

Very Important: When you multiply or divide both

sides of an inequality by a negative number, the

inequality sign must be reversed to remain true.

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6-8: Solve Two- Step Inequalities

Solve these just like two step equations. However, keep in mind that if you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality

sign. Example 1. What variable am I solving

for? x

3. What numbers are on that side that I need to get rid of to isolate the variable (get it alone)? 7 & -2

4. How can I get rid of those numbers using inverse operations? Subtract 7 and divide by -2.

5. There are no grouping symbols, what should I do?

Go to the next step

6. Undo addition or subtraction first. I must subtract 7 from both sides.

7. Undo multiplication and division next. I must divide both sides by -2. This means that I must reverse the inequality sign.

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Summary

1. Simplify each side of the equation 2. What variable am I solving for? n

4. What numbers are on that side that I need to get rid of to isolate the variable (get it alone)? 2/3 & 6

5. How can I get rid of those numbers using inverse operations? Subtract 6, divide by 2/3 or multiply by 3/2.

6. There are grouping symbols, what should I do? I can use the distributive property to simplify the expression and eliminate the parentheses.

7. Undo addition or subtraction first. I must subtract 4 from both sides now (instead of 6).

8. Undo multiplication and division next. I