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Chapter 3:Multi-step Equations and Inequalities by
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# Chapter 3:Multi-step Equations and Inequalities

## 3.1

### Words you should know

Vocaburaly Verbal Model:An expression that describes a problem using words as labels and using math sybols to relate the words Equation:A mathematical sentence formed by placing an equal sign,=,between two expressions Inverse operations: Two operations that undo each other

### Example 1:Using Subtraction and Division to Solve

Solving Two-Step Equations Example 1:Using Subtraction and Division to Solve First write normal eqautions: 3x+7= -5 Subtract 7 from each side because it is a constant number: 3x+7-7= -5-7 Then simplify: 3x= -12 Divide eachside by 3: 3x/3=-12/3 Simplify again and you get the answer: x= -4 To check your answer do the following Write the normal equations: 3x+7=-5 Substitute -4 instead of x, if you get the good answer bravo: 3(-4)+7?=-5 Check the solutions: -5=-5

### Example 2: Using Addition and Multiplication to Solve

Example 2: Using Addition and Multiplication to Solve Write original equation:x/2-3=1 Add 3 to each side :x/2-3+3=1+3 Simplify:x/2=4 Multiply each side by 2:2(x/2)=2(4) Simplify=8 To check your answer do the following Check: Write original equation:x/2-3=1 Substitute 8 for x:8/2-3?=1 Solution checks:1=1

### Example 3: Combining like terms after

Examaple 3 Write original equation:7-4y+=9 Subtract 7 from each side:7-4y-7=19-7 Simplify:-4y+12 Divide each side by 4:-4y/-4=12/-4 Simplify:y=-3 To check do the following: Write original equation:7-4y+=9 Substitute -3 for y:7-4(-3)?=19 Solution checks:19=19

## 3.2

Solving Equations Having Like Terms and Parentheses EXAMPLE 1:WRITING AND SOLVING AN EQUATION Subsitute:5.20n+0.80n+47=377 Combine like terms:6.00+47=377 Subract 47 from each side:6n+47-47=377-47 Simplify:6n=330 Divide each side by 6:6n/6=330/6 Simplify:n=55 EXAMPLE 2:Solving Equations Using The Distributive Property

### EXAMPLE 1:WRITING AND SOLVING AN EQUATION

EXAMPLE 1:WRITING AND SOLVING AN EQUATION Substitute:5.20n+0.80n+47=377 Combine like terms:6.00+47=377 Subtract 47 from each side:6n+47-47=377-47 Simplify:6n=330 Divide each side by 6:6n/6=330/6 Simplify:n=55

### EXAMPLE 2:Solving Equations Using The Distributive Property

Example 2   Write original equation:-21=7(3-x) Distributive property:-21=-21-7x Subtract 21 from each side:-21-21=21-7x-21 Simplify:-42=-7x Divide each side by -7:-42/-7=-7x/-7 Simplify:6=x

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## 3.3

### Solving an equation with variables on both sides

EXAMPLE 1   Write original equation:7n-5=10n+13 Subtract 7n from each side:7n-5-7n=10n+13-7n Simplify:-5=3n+13 Subract 13 from each side:-5-13=3n+13-13 Simplify:-18=3n Divivde each side by 3:18/3=3n/3 Simplify:-6=n

### Writing and solving equation

Example 2   Substitute:10s=5s+60 Subtract 5s from each side:10s-5s=5s-5s+60 Simplify:5s=60 Divide each side by 6:5s/5=60/5 Simplify:s=12

## 3.5

### Example 1 Solving inequality using division

Example 1 Original Inequality:-10t > 34 Divide each side by -10 .Reverse inequality symbol:-10t/-10 < 34/-10 Simplify:t

### Example 2 Solving an inequality using division

Example 2   Original Inequality:m/-3 >3 Multiply each side by -3 .Reverse inequality symbol:-3 times m/-3 < -3 times 3 Simplify:m

### Vocabulary

INEQUALITY: is a statement formed by placing an inequality symbol between two expressions. SOLUTION OF AN INEQUALITY: The sets  of all numbers that produce true statements when substituted for thr variable in the inequality EQUIVALENT INEQUALITIES: are inequalities that have the same solution

## 3.6

### Example 1 Writing and Solving a multi Step Inequality

Example 1 Substitute: 88+10g > 138 Subtract 88 from each side:88+10g-88 > 138-88 Simplify:10g >50 Divide each side by 10: 10g/10 >50/10 Simplify:g > 5

### Example 2: Solving a Multistep Inequality

Example 2   Original inequality:x/-4-6 > -5 Add 6 to each to side:x/-4-6+6 > -5+6 Simplify:x/-4> 1 Multiply each side by -4. Reverse inequality symbol.: -4xX/-4 < -4x1 Simplify: x < -4