Week 5: Equations and Inequalities
Solving Equations
-The goal is to get the variable alone!
-Use inverse operations to move terms to the opposite side of the equal sign.
- To undo addition – subtract
- To undo subtraction add
- To undo multiplication divide
- To undo division multiply
-
Examples:
1) x + 8 = 3
-8 -8subtract 8 from both sides
x = -5
Check:
-5 + 8 = 3
3 = 3
2) n – 5 = 2
+5 +5 add 5 to both sides
n = 7
Check
7 – 5 = 2
2 = 2
3) ½ n = 12
divide both sides by ½ multiply by reciprocal 2
n = 24
Check:
½ (24) = 12
12 = 12
4) 2n + 10 = 18
2n = 8 subtract 10 from both sides
n = 4 divide both sides by 2
Check
2(4) + 10 = 18
8 + 10 = 18
18 = 18
5) 3(2x – 1) = 15
6x – 3 = 15 distribute the 3
6x = 18 add 3 to both sides
x = 3 divide both sides by 6
Check:
3(2(3)- 1) = 15
3(6 – 1) = 15
3(5) = 15
15 = 15
6) 2m + 5 + 6m = 29
8m + 5 = 29 combine like terms
8m = 24 subtract 5 from both sides
m = 3 divide both sides by 8
Check:
2(3) + 5 + 6 (3) = 29
6 + 5 + 18 = 29
29 = 29
You Try:
Solve each equation. Check your work
1)n – 10 = 4
2)a + 12 = 7
3)-4n 28
4) = 5
5)
6)
7) 5x
8)
Equation: Word Problem Examples
#1. Mr. Lombard took his two children to a water park. He used paid $28 for an adult ticket. The price of admission for all three family members was $76. What was the price of each child’s ticket?
(price of adult ticket) + (number of children) (price of child’s ticket) = total cost.
#2. Lydia is saving money for her vacation. So far she has $82.50. Each week she sets aside 25% of her paycheck for the vacation. After 8 weeks, Lydia has $338.50 saved for vacation. What is the amount of Lydia’s weekly paycheck?
(amount already saved) + (number of weeks)(savings each week) = total amount saved
$82.50 + 8 ( 0.25x ) = $338.50
** The unknown was the paycheck
She was saving 25% of her paycheck
We write that as 0.25x
Her weekly paycheck was $128.
#3. Josh walked a total of 5 miles today. First he walked 1 mile from his house to his park. Then he walked laps around the trial at the park. Finally, he walked back home. How many laps did he walk around the trail?
I mile to park + 1 mile home + = 5 miles
1 + 1+
Josh walked 4 laps around the trail
You try:
- Scott was putting a fence around a rectangular garden next to his house. The length of the garden is 9 meters. A total of 21.5 meters of fencing is used. If w stands for the width of the garden in meters, which equation can be used to find its width?
- 2w + 9 = 21.5b. 2w + 18 = 21.5c. 2w – 21.5 = 9d. 2w + 21.5 = 18
- If 10 is first subtracted from both sides of the equation , What do you do next to solve for x?
- Subtract from both sides of the equation
- Multiply by x and by 15
- Subtract from both sides of the equation
- Multiply and by 15
- Lane stacks some books in a storage crate that weighs 5 pounds. Each book weighs 1 pounds. The total weight of the crate with all three books in it is 53 pounds. Which equation CANNOT be used to find n, the number of books in the crate?
- 1 b. 1.375n + 5 =53c. 1.375n = 48
- For babysitting, Nicole charges a flat fee of $10, plus $5 per hour. She earned $27.50 on Tuesday night. Write an equation to determine how many hours Nicole babysat on Tuesday night.
Equation: ______
Solution: ______
- Banners at the school store were on sale for $3 off the regular price. Louis bought 4 banners on sale and paid a total of $18. Write and solve an equation to find the regular price of one banner.
Equation: ______
Solution: ______
- The length of each side of the two congruent sides of an isosceles triangle is 2x + 3. The length of the third side is 2x. Its perimeter is 36 centimeters. Write an equation that could be used to find the value of x. Solve for x and then find the length of all three sides.
Equation: ______
Side 1: ______Side 2: ______Side 3: ______
- The Hair Care Salon charges a stylist $30 per day to rent a station at the salon. Rhonda, a sylist, makes $10.50 on each haircut. Write an to equation help her decide how many haircuts she must give in one day to make $138 after paying rent for her station? Then solve the equation.
Equation: ______
Solution: ______
Solving Problems with Inequalities
Solving inequalities is similar to solving equations. But when you multiply or divide each side of an inequality by a negative number, you reverse the inequality symbol.
: Less than
less than or equal to. AT MOST
: greater than
: greater than or equal to AT LEAST
Examples
#1. Mr. Thomas brings $100 to a fundraiser. He wants to leave the event with at least $50 in his pocket. Guests at the fundraiser buy raffle tickets for several different prizes. Each raffle ticket costs $2.50. How many raffle tickets can Mr. Thomas buy and still leave with at least $50 in his pocket?
(starting amount) – (ticket price) * (number of tickets) is greater than or equal to the (amount left)
100 - 2.50 *t 50
100 – 2.5t 50
-100 -100
- 2.5t- 50 Reverse the symbol because we are dividing by a negative number
- 2.5 -2.5
t 20 He can buy 20 or fewer tickets
#2. Chang wants to spend at most $60 on socks and sneakers. He finds a pair of sneakers that he likes for $36. If socks are $3 per pair, how many pairs could Chang buy?
Price of sneakers + 3 * p pairs of socks must be less than or equal to $60
36 + 3p 60
-36 -36
3p 24
3 3
p 8 He can buy 8 or less pairs of socks.
#3. Mrs. Sanchez is building a laundry room in the basement of the apartment building she owns. Given the layout of the basement, she wants to width of the room to be 20 feet and the length to be longer than the width. If she wants the area of the room to be more than 500 square feet, what could be the length?
The product of the width and length must be greater than 500 square feet.
Look at this diagram
The length can be anything greater than 5 ft.
#4. At the beginning of baseball season, Coach Thorne takes inventory of the team equipment to see what he needs. He counts 24 baseballs, but he needs to start off the season with at least 100 balls. The balls that he uses are sold in packages of 4. How many packages could the coach buy?
24 + 4x 100
-24 -24
4x 76
4 4
x 19Coach Thorne can buy 19 or more packages
You Try:
#1. What is the solution to the inequality 35 – 2x 10?
- x -12.5
- x -12.5
- x
- x 12.5
2. The town recreation center is sponsoring a charity drive. The goal is to raise at least $2,500. So far, $2,200 has been collected. If there are 5 days left in the charity drive, how much money on average should they aim to collect each day?
a. x 500
b. x 440
c. x 60
d. x
3. Market and More is having a cereal sale. Every box of cereal is $0.60 off the regular price. Jane has at most $10 to spend and she wants to buy 4 boxes of cereal. Write an inequality to determine the regular prices of boxes of cereal that she can afford. Solve the inequality and explain what the solution means.
4. The dance committee has a budget of $125 to decorate the gym for the spring dance. They have already spent $65. Some members want to buy helium balloons that cost $0.80 each. Write and solve an inequality to show the number of balloons that the dance committee could buy.
Show your work.
Answer The committee can buy ______balloons
#5. Rocky pays a flat rate of $5 per month plus $0.10 per minute for his long-distance phone plan. He wants his total monthly bill to be less than $20.
Part A
Write an inequality that represent the maximum number of minutes, m, Rocky can talk to stay within his budget.
Answer: ______
Part B
Solve that inequality to find the maximum number of minutes Rocky can talk to stay within his budget.
Show your work.
Answer: ______
Part C: On a number line, draw a graph that represents the number of minute Rocky can talk to stay within his
budget.
Graphing Inequalities on a Number Line:
If the variable comes after the inequality symbol, switch the direction
3 > n is the same as n < 3
From 2013 State Test:
Extra Practice:
- Mrs. Williams is 52 years old and her daughter is x years old. Mrs. William’s age is 6 years more than twice her daughter’s age. Which of the following equations can be used to find her daughter’s age?
- Mike and John helped Mr. Peterson repair the roof of his house and earned $x. After they divided the money equally, they each received $225. Which of the following equations can be used to determine x?
- The poster below shows the charges a rental company charges for a motorcycle.
Motorcycle Rental Charges
One Time Charge--$13
Each Day--$6
Which of the following expressions represents the total rental cost, in dollars, of 1 motorcycle for b days?
- Ten more than Rosanne’s average typing speed, t, is greater than 45. What values of t will make the above statement true?
- A bucket can hold 30 liters of water. S represents the liters of water can be poured into the bucket. Which of these inequalities shows the water poured into the bucket if it already contains 7 liters of water?
Combine the like terms in the expression. Write your answer in the space provided.
1.2.
Find the sum or difference. Write your answer in the space provided.
3. 4.
5. 6.
Factor out the coefficient of the variable. Write your answer in the space provided.
7. 8.
9.Write an expression in simplest form that represents the perimeter
of the polygon.
Solve the equation. Check your solution.Circle your final answer.
11. 12. 13.
14. 15. 16.
Write the word sentence as an equation. Then solve.
17.11 more than a number q is negative 15.18. The difference of a number m and 30 is 10.
19.One-third of a number t is equal to 7.20.The quotient of 5 plus a number d and negative 2 is 14.
In Exercises 21-24, write an equation. Then solve.
21.The temperature is A high pressure front increases the temperature to By how many degrees did the temperature increase?
22.The monthly dues for a premium membership at a health club is $15 more than the cost of a standard membership. The premium membership is $40 per month. What is the cost of a standard membership?
23.A pack of cardinal flower seeds costs $4, and a pack of petunia flower seeds costs $2.50. You buy the same number of packs of each type of flower and spend $39. How many packs of each do
you buy?
24.An egg carton holds 12 eggs. A breakfast buffet uses 96 eggs by 8 a.m. When the buffet ends at 10:30 a.m., a total of 156 eggs were used. How many cartons of eggs were used after 8 a.m.?