Topic 1. Solving Equations and Inequalities 1. Solve the following equation

Transcription

1 Topic 1. Solving Equations and Inequalities 1. Solve the following equation Algebraically 2( x 3) = 12 Graphically 2( x 3) = Solve the following equations algebraically a. 5w 15 2w = 2(w 5) b (t ) = 1 4 Inequalities Remember when graphing inequalities I do 1. Solve and graph the given inequality. a. - 4m 7 11 b. 7 2 x < x Solve the inequality below to determine and state the smallest possible value for x in the solution set. 3(x + 3) 5x 3 3. *** Given 4x + ax 8 > 12, determine the largest integer value of a when x = 3.

2 REGENTS PROBLEMS 1. The inequality is equivalent to 2. What is the value of x in the equation? a. 4 c. 8 b. 6 d Which value of x satisfies the equation? a c b d Given, determine the largest integer value of a when. 5. Solve the inequality below to determine and state the smallest possible value for x in the solution set. 6. Solve for x algebraically: If x is a number in the interval determined these values., state all integers that satisfy the given inequality. Explain how you 7. Determine the smallest integer that makes true.

3 Topic 2. Literal Equations (Rearranging Formulas) Formula: A formula is an equation that shows the relationship between two or more variables. Transform: To transform something is to change its form or appearance. In mathematical equations, a transformaion changes form and appearance, but does not change the relationships between variables. To transform a formula or equation usually means to isolate a specific variable. Literal Equations - Solve each equation for the indicated variable. Solve for r Solve for the height h 4 V r 3, for r 3 A = 1 bh, for h 2 Regents Problems 1. The equation for the volume of a cylinder is. The positive value of r, in terms of h and V, is 2. The formula for the volume of a cone is. The radius, r, of the cone may be expressed as

4 3. The distance a free falling object has traveled can be modeled by the equation, where a is acceleration due to gravity and t is the amount of time the object has fallen. What is t in terms of a and d? 4. The formula for the area of a trapezoid is. a. Express in terms of A, h, and. b. The area of a trapezoid is 60 square feet, its height is 6 ft, and one base is 12 ft. Find the number of feet in the other base. 5. The volume of a large can of tuna fish can be calculated using the formula. a. Write an equation to find the radius, r, in terms of V and h. b. Determine the diameter, to the nearest inch, of a large can of tuna fish that has a volume of 66 cubic inches and a height of 3.3 inches.

5 Topic 3. Solve Systems of Equations and Inequalities by Graphing On the set of axes below, draw the graph of the equation y = 3 x Is the point (3,2) a solution to the equation? Explain your answer based on the graph drawn. Determine the slope and y - intercepts of the graph of the equation 9x 27y = Write an equation for the line that passes through the points (1, 6) and (5, 2). When Graphing Linear Inequalities: Solve for the variable y and read the inequality from y Remember that if you multiply or divide by a negative the inequality sign switches If you are graphing > or < use a dotted boundary line to show any point on the line is not in the solution set Solve for the variable y Remember If you that are if graphing you multiply or or use divide a solid by a boundary negative line to show any point on the line is included in the the inequality sign switches solution set. Shade ABOVE when: When solving Systems of Equations we find the point of intersection given two functions Shade BELOW when:. Solving by graphing - Remember to *** label*** and STATE THE SOLUTION(S) 1. Graph the solution to 3y 4x > 4. Circle all ordered pairs that are solutions (0, 4) (0, 1) (3, 0) (4, -5) (-2, 4) (0, 2) (6, 4)

6 You Try 2. Graph the solution to 2x 3y < Graph the solution to x y 2 4. On the set of axes below, graph the inequality.

7 REGENTS PROBLEMS 1. What is one point that lies in the solution set of the system of inequalities graphed below? 2. Given: Which graph shows the solution of the given set of inequalities? 3. Which ordered pair is not in the solution set of and?

8 4. Which graph represents the solution of and? 5. Which inequality is represented in the graph below? 6. The graph of an inequality is shown at right. a. Write the inequality represented by the graph. b. On the same set of axes, graph the inequality. c. The two inequalities graphed on the set of axes form a system. Oscar thinks that the point is in the solution set for this system of inequalities. Determine and state whether you agree with Oscar. Explain your reasoning.

9 Topic 4. Applications of systems of Inequalities (Word Problems) **Every word problem is different. Write a few phrases that you will come across, please read carefully and pay attention to the direction of the inequality symbol. We do 1. Edith babysits for x hours a week after school at a job that pays $4 an hour. She has accepted a job that pays $8 an hour as a library assistant working y hours a week. She will work both jobs. She is able to work no more than 15 hours a week, due to school commitments. Edith wants to earn at least $80 a week, working a combination of both jobs. Write a system of inequalities that can be used to represent the situation. Graph these inequalities on the set of axes below. Determine and state one combination of hours that Will allow Edith to earn at least $80 per week while working no more than 15 hours.

10 Now You Try 2. Leah is buying clothes for spring break! She has a few outfits already, but definitely needs a few more! She is choosing from a bunch of shirts, x that cost $8 each and skirts, y that cost $12 each. She can spend at most $192. Leah does need to buy at least 5 items to fill her suitcase. a) Write a system of inequalities to model this situation. 20 b) Graph the inequalities on the grid provided. 10 c) Name a combination of shirts and skirts that will give her at least 5 items, but cost no more than $ David has two jobs. He earns $8 per hour babysitting his neighbor s children and he earns $11 per hour working at the coffee shop. Write an inequality to represent the number of hours, x, babysitting and the number of hours, y, working at the coffee shop that David will need to work to earn a minimum of $200. David worked 15 hours at the coffee shop. Use the inequality to find the number of full hours he must babysit to reach his goal of $ A high school drama club is putting on their annual theater production. There is a maximum of 800 tickets for the show. The costs of the tickets are $6 before the day of the show and $9 on the day of the show. To meet the expenses of the show, the club must sell at least $5,000 worth of tickets. a) Write a system of inequalities that represent this situation. b) The club sells 440 tickets before the day of the show. Is it possible to sell enough additional tickets on the day of the show to at least meet the expenses of the show? Justify your answer.

11 REGENTS PROBLEMS 1. The cost of a pack of chewing gum in a vending machine is $0.75. The cost of a bottle of juice in the same machine is $1.25. Julia has $22.00 to spend on chewing gum and bottles of juice for her team and she must buy seven packs of chewing gum. If b represents the number of bottles of juice, which inequality represents the maximum number of bottles she can buy? 2. John has four more nickels than dimes in his pocket, for a total of $1.25. Which equation could be used to determine the number of dimes, x, in his pocket? 3. A cell phone company charges $60.00 a month for up to 1 gigabyte of data. The cost of additional data is $0.05 per megabyte. If d represents the number of additional megabytes used and c represents the total charges at the end of the month, which linear equation can be used to determine a user's monthly bill? 4. Connor wants to attend the town carnival. The price of admission to the carnival is $4.50, and each ride costs an additional 79 cents. If he can spend at most $16.00 at the carnival, which inequality can be used to solve for r, the number of rides Connor can go on, and what is the maximum number of rides he can go on? a. ; 3 rides c. ; 14 rides b. ; 4 rides d. ; 15 rides 5. In 2013, the United States Postal Service charged $0.46 to mail a letter weighing up to 1 oz. and $0.20 per ounce for each additional ounce. Which function would determine the cost, in dollars,, of mailing a letter weighing z ounces where z is an integer greater than 1? 1) 2) 3) 4) 6. Alex is selling tickets to a school play. An adult ticket costs $6.50 and a student ticket costs $4.00. Alex sells x adult tickets and 12 student tickets. Write a function,, to represent how much money Alex collected from selling tickets.

12 7. Jackson is starting an exercise program. The first day he will spend 30 minutes on a treadmill. He will increase his time on the treadmill by 2 minutes each day. Write an equation for, the time, in minutes, on the treadmill on day d. Find, the minutes he will spend on the treadmill on day Each day Toni records the height of a plant for her science lab. Her data are shown in the table below. The plant continues to grow at a constant daily rate. Write an equation to represent, the height of the plant on the nth day

13 Topic 5. Systems of Equations Algebraically Elimination is a method of solving systems of equations. Elimination allows us to combine the equations into one in which there is only one variable. Steps for Solving Systems of Equations by Eliminations Step 1: Line up the equations and variables vertically. Step 2: Find two variables that have opposite coefficients. IF there are no variables with opposite coefficients, use multiplication to create opposite coefficients. Step 3: Add the equations together and see that the variable with the opposite coefficients is eliminated. Step 4: Solve for the remaining variable. Step 5: Use that solution and one of the origin equations to solve for the eliminated variable. Step 6: Check your solutions in BOTH ORIGINAL equations and write the solution as a coordinate pair. Examples: 5x 6y = { x + 2y = 16 x + 2y = 9 2. { 2y = 1 x Example 3: Lulu tells her little brother Jack that she is holding 20 coins all of which are dimes and quarters. They have a value of $4.10. She says she will give him the coins if he can tell her how many of each she is holding. Help Jack earn the money and determine the number of dimes and number of quarters Lulu is holding.

14 REGENTS PROBLEMS 1. Which pair of equations could not be used to solve the following equations for x and y? a) c) b) d) 2. Last week, a candle store received $ for selling 20 candles. Small candles sell for $10.98 and large candles sell for $ How many large candles did the store sell? 3. Mo's farm stand sold a total of 165 pounds of apples and peaches. She sold apples for $1.75 per pound and peaches for $2.50 per pound. If she made $337.50, how many pounds of peaches did she sell? a. 11 c. 65 b. 18 d An animal shelter spends $2.35 per day to care for each cat and $5.50 per day to care for each dog. Pat noticed that the shelter spent $89.50 caring for cats and dogs on Wednesday. a. Write an equation to represent the possible numbers of cats and dogs that could have been at the shelter on Wednesday. b. Pat said that there might have been 8 cats and 14 dogs at the shelter on Wednesday. Are Pat s numbers possible? Use your equation to justify your answer. c. Later, Pat found a record showing that there were a total of 22 cats and dogs at the shelter on Wednesday. How many cats were at the shelter on Wednesday?