Warm Up

• Write down objective and homework in agenda

• Lay out homework (Kuta Inequalities, any 18)• Homework (Review Sheet!)

Unit 1 Common Core Standards• 8.EE.7 Solve linear equations in one variable.• a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or

no solutions. Show which of these possibilities is the case by successively transforming the given• equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results

(where a and b are different numbers).• b. Solve linear equations with rational number coefficients, including equations whose solutions

require expanding expressions using the distributive property and collecting like terms.• 8.G.6 Explain a proof of the Pythagorean Theorem and its converse.• 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-

world and mathematical problems in two and three dimensions.• 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate

system.• A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include• equations arising from linear and quadratic functions, and simple rational and exponential

functions.• Note: At this level, focus on linear and exponential functions.• A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in

solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R. Note: At this level, limit to formulas that are linear in the variable of interest, or to formulas involving squared or cubed variables.

Unit 1 Common Core Standards• A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous

step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

• A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Note: At this level, focus on linear and exponential functions.

• A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.• A-SSE.1 Interpret expressions that represent a quantity in terms of its context.• a. Interpret parts of an expression, such as terms, factors, and coefficients.• b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For• example, interpret P(1+r)n as the product of P and a factor not depending on P.• Note: At this level, limit to linear expressions, exponential expressions with integer exponents and• quadratic expressions.• G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a• circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and• informal limit arguments.• Note: Informal limit arguments are not the intent at this level.• G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*• Note: At this level, formulas for pyramids, cones and spheres will be given.• G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles,• e.g., using the distance formula.

Unit 1 Common Core Standards• N-Q.1 Use units as a way to understand problems and to guide the solution of multi-

step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

• N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

• N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (5 1/3)3 must equal 5.

• N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

• Note: At this level, focus on fractional exponents with a numerator of 1.• MP.1 Make sense of problems and persevere in solving them.• MP.2 Reason abstractly and quantitatively.• MP.4 Model with mathematics.• MP.7 Look for and make use of structure.

Warm Up

Inequality Word Problems

• Same as equations, define a variable, write an inequality, and solve

AT MOST MEANS ≤AT LEAST MEANS ≥

Real –World Problem SolvingJulie has $500 in her banking account. She writes checks for $240.78 and $150.50. If she does not want her account balance to fall below $50, how much money can Julie spend?

m = money Julie can spend 240.78 + 150.50 + m + 50 ≤ 500240.78 + 150.50 + m ≤ 450391.28 + m ≤ 450391.28 ─ 391.28 + m ≤ 450 ─ 391.28m ≤ $58.72

Example

• The track team at Hale High School plans to sell t-shirts to raise money for new equipment. At Ted’s Tees, printing costs are $0.80 per shirt and the cost for each T-shirt is $3.75. The shop also charges a $125 fee per order for the silk-screen design.

• In order for the track team to make a profit, the income from selling the T-shirts must be greater than the cost of making the T-shirts.

• Write an inequality comparing the two expressions from part a and b.

• x = number of shirts• Income > cost• 3.75x > 0.8x + 125

Example

• 3.75x > 0.8x + 125• Solve the inequality. How many shirts must

they sell to make a profit?• x > 42.37388…• You must sell more than 43 shirts or x > 43

shirts

Word Problems

• r = number of rides

• Therefore, Carlos can go on 25 rides.

5 0.80 25

0.80 20

25

r

r

r

Word Problems

• s = number of suckers bought • Therefore, the minimum number of suckers bought

was 48.• This is true because of the “at least.” If the

problem had said Linus had more suckers than Nicholas then the inequality symbol would not have included the equal part and the answer would have been 49. 1 1

5 32 31 1

82 31

86

48

s s

s s

s

s

• p = number of pink post-its • Therefore, the most pink post-its Annah

could have is 11. • Since you don’t usually count parts of post-it

notes and she has to have fewer than 12, the answer is 11.

4 3 2 120

10 120

12

p p p p

p

p

• I ordered four new CDs by mail. Each costs the same amount and there was an additional $5.00 shipping/handling charge that brought the total to $33.00. Write and solve an equation to find the cost of one CD.

• $7

• t = number of extra toppings • Therefore, Silvia purchased pizzas with no more than 3

extra toppings each.• You can’t have a part of a topping (no half the pizza this

way half another in this problem!) so the answer has to be 3, 2, or 1. Since the question asked what was the most she could have, the answer must be 3.

3(12 0.90 ) 45

36 2.70 45

2.70 9

3.333

t

t

t

t

You Try!

• Trevor is a salesperson who is paid a salary of $500 plus 2% commission. Write and solve an equation to determine how much Trevor must sell to earn $2,000 this month.

• $75, 000

More Practice

• 1. To make a B in Mr. McLean’s class, students must have at least a total of 255 points. Sean made a 91 on his first assignment and an 80 on his second assignment. What must he score on his third assignment if he wants to have a B?

• 2. Phil’s truck will hold 2100 pounds. If he has 40 bags of horse feed which weigh 50 pounds each, how many pounds of hay could be added to the truck?

• 3. An auto shop estimates it will cost no more than $250 for parts and labor to repair a car. If the parts cost $72.00, then labor must cost no more than what amount?

Answers

• 1. 91 + 80 + g ≥ 255; g ≥ 84 score• 2. (40)(50) + h ≤ 2100; h ≤ 100 pounds• 3. 72 + x ≤ 250; x ≤ $178

You Try!

• Maggie is selling Girl Scout cookies. This week Ms. Coleman bought 3 boxes, Mr. Papile bought 4 boxes, and her friend Carroll bought 1box. Mr. Somma cannot eat cookies so he gave a $7 donation to Maggie’s troop. If Maggie sold 10 boxes last week and the amount of money she collected last week equals the amount of money she collected this week, write and solve an equation to determine how much each box of cookies costs.

• $3.50

More Examples!

• The Smiths are planning their daughter’s wedding. They will pay $225 to rent a reception hall for the evening. The cost to the reception hall per plate for dinner is $6.50, and the reception hall is charging the Smith’s $15.00 per plate. If the reception hall will earn a profit of $1,287.50 for the wedding, how many attendees will be present? Assume that the cost to rent the hall is entirely profit.

Answer• number of attendees = 125• Let x be the number of attendees and p be the

profit.• hall rental + 15x – 6.5x = p

More Examples!

• The total bill for new dirt and fencing at Clovernook baseball field is $378.00. The cost of new fencing is $270.00. If the cost of new dirt is $4.50 for a cubic foot, how many cubic feet of dirt were used for the field? Show all work.

Answer

• Let x represent the number of cubic feet used.

Extra Help

http://www.regentsprep.org/Regents/math/ALGEBRA/AE8/PSolvIn.htmhttp://www.regentsprep.org/Regents/math/ALGEBRA/AE8/PracIneqD.htm