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# Chapter 3 -Multi-step equations and inequalities

This year we began learning about solving Multi-step equations and inequalities. We have already learned how to solve two-step equations, how to solve equations having Like terms and parentheses and how to solve equations with variables on both sides. Today (November 10 2011) we are going to learn how to solve inequalities using addition or subtraction.

## 3.1 Solving Two-Step Equations

3.1 Includes many types of solving, solving two step equations. Solving two- step Equations .... Using Subtraction and division to solve.. Using Addition and Multiplication to solve. Using an Equation with Negative Coefficients. The last one is Writing and Solving a two-step Equation

### Using Subtraction and Division to solve

Solve 3x + 7= -5 3x + 7 = -5                    Write original equation 3x +  7 -  7 =-5 - 7          Subtract 7 from each side 3x=-12                          Simplify 3x/3 = -12/3                   Divide each side by 3 x=-4                              simplify Answer: The solution is 4 This was one example of Using subtraction and Division to solve.

### Using Addition and Multiplication to solve

Solve x/2 - 3 = 1 x/2 - 3 = 1                        Write original equation x/2 - 3 + 3 = 1 + 3            Add 3 to each side x/2 = 4                            Simplify 2 (x/2) = 2(4)                    Multiply each side by 2 x=8                                 Simplify Answer: The solution is 8 This was one example of Solving addition and Multiplication.

### Solving an Equation with Negative Coefficients

Solve 7 -4y = 19 7-4y = 19                       Write original equation 7 - 4y - 7 = 19 - 7            Subtract 7 from each side -4y = 12                         Simplify -4y/-4 = 12/-4                  Divide each side by -4 y= -3                              Simplify Answer: The solution is -3 This was one example of solving an equation with Negative Coefficients.

### Writing and Solving a two-step equation

The problem is "You are buying a drum set that costs \$495. The music store lets you make a down payment. You can pay the remaining cost in three equal monthly payments with no interest charged. You make a down payment of \$150. How much is each monthly payment? The example I found on book is about this problem, but the question/the thing you need to find is totally different. The question is: "Find the monthly payment for the drum set." Solution "Let p represent the monthly payment. Write a verbal model. To solve that kind of problem, you should do this: "Total cost of drum set = amount of down payment + Number of months x monthly payment. 495= 150 + 3p Substitute 495 - 150 = 150 + 3p - 150 Subtract 150 from each side 345= 3p Simplify 345/3 = 3p/3 Divide each side by 3 115 = p simplify Answer: The monthly payment is \$115

## 3.2 Solving Equations having Like terms and Parentheses

3.2 is mostly about distributive property and to combine like terms when you distribute.

### Solving Equations using the distributive property

Solve : -21 = 7(3 - x) -21 = 7(3 - x)                      Original equation -21 = 21 - 7x                      Distributive property -21 - 21 = -21 - 7x - 21        Subtract 21 from each side -42= -7x                             Simplify -42/-7 = -7x/7                      Divide each side by -7 6 = x                                  Simplify This was one example of solving equations using the distributive property.

### Combining like terms after distributing

Solve: 5x - 2(x - 1) = 8 5x - 2(x - 1) = 8        Original Equation 5x - 2x + 2 = 8         Distributive Property 3x + 2 = 8               Combine like terms 3x + 2 - 2 = 8 - 2     Subtract 2 from each side 3x = 6                    Simplify 3x/3 = 6/3              Divide each side by 3 x = 2                     Simplify Answer: The solution is 3

## 3.3 Solving Equations with variables on both sides

### Solving an Equation with the variable on both sides

Solve: 7n - 5 = 10n + 13 7n - 5 = 10n + 13                    Write original equation 7n - 5 - 7n = 10n + 13 - 7n      Subtract 7n from each side -5  =  3n + 13             Simplify -5 - 13 = 3n + 13 - 13             Subtract 13 from each side - 18 = 3n                           Simplify -18n/3 = 3n/3                         Divide each side by 3 -6 = n                                    Simplify   Answer: The solution is -6

### Writing and Solving an Equation

The problem: The Spanish club is arranging a trip to a Mexican restaurant in a nearby city. Those who must go share the \$60 cost of using a school bus for the trip. The restaurant's buffet costs \$5 per person. How many students sign up for this trip in order to limit the cost to \$10 per student? The question:  How many students must go on the Spanish club trip to the Mexican restaurant in order for the cost per person to be \$10? Solution: Let s represent the number of students. Write a verbal model. The method to solve that problem is to do that: cost per student x number of students = cost of buffet x numebr of students + cost of school bus. 0s  = 5s + 60                  Substitute 10s  -  5 = 5s - 5s + 60          Subtract 5s from each side 5s = 60                                Simplify 5s/5 = 60/5                          Divide each side by 5 s = 12                                 Simplify Answer: The club needs 12 students to go on the trip.

### An Equation with no solution

Solve : 5(2x + 1) = 10x 5(2x+1) = 10x                              Original equation 10x + 5 = 10x                              Distributive property 10x + 5 - 10x = 10x - 10x             Subtract 10x from each side 5 = 0                                          Simplify The statement 5 = 0 is not true, so the equation has no solution.

### Solving an equation with all numbers as solutions

Solve:  6x + 2 = 2(3x + 1) 6x + 2 = 2(3x + 1)     Original Equation 6x + 2 = 6x + 2         Distributive Property The statement 6x + 2 = 6x + 2 is true, the equation has every number as a solution.

### Solving an Equation to find a perimeter

Find the perimeter of the square. 1) A square has four sides of equal length. Write an Equation and solve for x. 2x = x + 4                       Original equation 2x - x = x + 4 - x             Subtract x from each side x = 4                              Simplify 2) Find the length of one side by substituting 4 for x in either expression. 2x = 2(4) = 8                  Substitute 4 for x and multiply 3) To find the perimeter, multiply the length of one side by 4. 4 x 8 = 32 Answer: The perimeter of the square is 32 units.

## Where I got my explanations

I got the explanations from the Math Text Book.

## 3.4 Solving inequalities using addition or subtraction

### Writing and graphing an Inequality

Problem: The freezing point of water is 0 degrees Celsius. At temperature at or below the freezing point, water is a solid (ice). Write an inequality that gives the temperatures at which water is a solid. Solution: Let t represent the temperature of water. Water is a solid at temperature less than or equal to 0 degrees celsius. Answer: The inequality is all numbers less than 0 or equal to t .

### Solving an Inequality using subtraction

Solve: m + 5 > (or equal to) 10 m + 5 > (or equal to) 10                    Original inequality m + 5 - 5 > (or equal to) 10 - 5            Subtract 5 from each side m > (or equal to) 5                             Simplify   Answer: The solution is m > (or equal to) 5

### Solving an Inequality using addition

Solve: -10 > x - 12 -10 > x - 12 Original Inequality -10 + 12 > x - 12 + 12 Add 12 to each side 2 > x Simplify Answer: The solution is 2> x

### Writing and Solving an Inequality

You are competing in a triathlon, a sports competition with three events. Last year, you finished the triathlon in 85 minutes. The table shows show's your times for this year's two events. What possible times can you post in the running event and still beat last year's finishing time?   Triathlon times Event               Time (min) Swimming               17 min Biking                     45 min Running                      ?   Solution: Let t represent this year's running time. Write a verbal model. The method is : Swimming time + biking time + running time < last year's finishing time 17 + 45 + t < 85                                 Substitute 62  + t < 85                                        Simplify 62 + t - 62 < 85 - 62                           Subtract 62 from each side t < 23                                                Simplify   Answer: To beat last year's finishing time, you must post a time in the running event that is less than 23 minutes.

## 3.5 Solving Inequalities using multiplication or division

### Solving an Inequality using multiplication

Solve: m/ -3 > 3 m/-3 > 3                         Original Inequality -3 x m/-3 < -3 x 3            Multiply each side by -3. Reverse Inequality symbol m < -9                           Simplify

### Solving an Inequality using division

Solve: -10t > (or equal to) 34 -10t > (or equal to) 34               Original equation -10t/-10t < (or equal to) 34/-10   Divide each side by -10. Reverse inequality symbol t < (or equal to) -3.4                 Simplify

### Writing and solving an Inequality

Problem: Some flocks of Canada geese can fly nonstop for up to 16 hours. In this time, a flock can migrate as a far as 848 miles. At what average speeds can such a flock fly during migration?   Find the average speeds at which the flock of Canada geese. Solution: Let s represent the average flight speeds. Write a verbal model. This is the method to solve this problem: Flight time x Average flight speeds < (or equal to) maximum flight distance   16s < (or equal to) 848                      Substitute 16s/16 < (or equal to) 848/16             Divide each side by 16 s < (or equal to)  53                          Simplify   Answer: The flock of Canada geese can fly at average speeds of 53 miles per hour or less during migration.

## 3.6 Solving Multi-step Inequalities

### Writing and Solving a Multi-Step Inequality

Problem: Your school's soccer team is trying to break the school record for goals scored in one season. Your team has already scored 88 goals this season. The record is 138 goals. With 10 games remaining on the schedule, how many goals, on average, does your team need to score per game to break the record? Find the average number of goals your team needs to score per game to break the school record. Solution: Let g represent the average number of goals scored per game. Write a verbal model.   The method to solve that problem: Goals scored this season + number of games left x goals scored per game > school record 88 + 10g > 138                     Substitute 88 _ 10g - 88 > 138 - 88        Subtract 88 from each side 10g > 50                              Simplify 10g/10 > 50/10                     Divide each side by 10 g > 5                                    Simplify   Answer: Your team must score, on average, more than 5 goals per game