ACT Elementary Algebra Review Contents Simplifying Expressions... Review... Practice Problems... FOIL and Multiplying Binomials... 4 Review... 4 Practice Problems... 4 Inequalities... 7 Review... 7 Practice Problems... 9 Translating word problems into algebra... 10 Review... 10 Practice Problems... 10 Factoring basic equations... 1 1
Simplifying Expressions Review There are several questions on the ACT that require you to simplify algebraic expressions. Here s an overview of the terminology. Like terms in an expression are those that have the same variables with the same exponents. The coefficients may differ. Example: x - 6n + 4 + 3x + 9xn - 1 + 3n - n + 1x 3x 3 x and 1x are like terms -6n and -n are like terms 4 and -1 are like terms Combining like terms means to add or subtract the like terms in the expression. On the ACT, you will usually be told to simplify the expression. The implication is that you will simplify by combining like terms via addition or subtraction. Example: Simplify x y + 3 xy + 7y - 8 + 1x (x + 1x) + (-y + 7y) + (3-8) xy (grouping like terms) = 3x + 5y - 5 xy (add/subtract) When an expression has grouping symbols, you need to simplify them following the order of operations. Example: Simplify 4(3m - 4n + 3) - (3m + 3n ) - (5m - 13m + 8n ) 1m - 16n + 1-3m - 3n - 10m + 6m - 16n (first need to distribute) (1m - 10m) + (-3m + 6m ) + (-3n - 16n ) - 16n + 1 (group like terms) m + 3m - 19n - 16n + 1 (add/subtract) Practice Problems 1. Simplify 4a - 5.7b + 3.3c - 1.4a +.b - 5c. Simplify x - 3( 1 3 + 6x + y) + ( x + 6y - 1) 3. Simplify [x - (3 - x)] 4. Simplify 9(x - 5) - 3(8 + 4x) + 3[x - (4 + x)]
5. Simplify 4[3(x + 5xy + 3y ) - (x + 4y )] - [1xy - (6x - xy - 3y )] 6. If 7 + 3x =, what is x? A. B. 5 C. 10 D. E. 14 7. The product ( y)(3 ) is equal to: F. 5 G. 6 H.6 J. 6 K. 15 8. What polynomial must be added to - x + 6 so that the sum is 3 + 8x? A. 4 + 5x - 6 B. 4-10x + 6 C. 3-10x + 6 D. + 10x - 6 E. - x + 6 Answer Key: 1. -8.4a - 3.5b - 1.7c. -17x - 3. 4x - 6 4. -3x -81 5. 4x + 46xy + 14y 6. C 7. G 8. D 3
FOIL and Multiplying Binomials Review The FOIL Method Simplify (x-)(3x+5) When we have the product of two binomials, the FOIL procedure tells us we can: 1.) Multiply the First terms: x and 3x to obtain 6x..) Multiply the Outer terms: x and 5 to obtain 10 x 3.) Multiply the Inner terms: - and 3x to obtain -6x 4.) Multiply the Last terms: - and 5 to obtain -10 Then simplify the resulting polynomial if possible: O I ( x )( 3x + 5) + 4x 10 = 6x + 10x 6x 10 = 6x F O I L F L Special Products: The ACT people love to ask about three special kinds of binomial products: (x + y) (x y) = - (x + y) (x + y) = + xy + (x y) (x - y) = - xy + Practice Problems 1. x y =? A. x - xy + y 4
B. x + xy - y C. x - xy - y D. x - xy + y E. x - y. If two less than a certain number is multiplied by three more than the number, the product is one less than the square of the number. Find the number. A. 3 B. 4 C. 5 D. 6 E. 7 5
3. As shown in the figure below, a square picture is surrounded by a rectangular frame that measures inches wide. If the area of the frame is 48 square inches, find the dimensions of the picture in inches. F. 4 G. 1 H. 14 J. 16 K. 0 in x in Answer key: 1. A. x y x y x y = x - xy - xy + - xy +. C (x ) (x + 3) = x - 1 x + 3x x 6 = x - 1 x + x 6 = x - 1 x = 5 3. J (Hint: The area of the whole thing equals the area of the picture plus the area of the frame.) Area of whole thing = x (area of inner square) + 48 (area of frame) What else do we know? Area of whole thing = whole side x whole side = (x + in border + inch border) (x + in border + inch border) = (x + 4) (x + 4) (x + 4) (x + 4) = x + 48 x + 8x + 16 = x + 48 8x = 3 x = 4. 6
So the area of the inner square is x = 4 = 16 Inequalities Review On the ACT, there may be one or two math problems involving inequalities. Usually, questions about inequalities only involve one variable and do not involve graphing. Infrequently, however, there will be an inequalities problem with two variables and/or that may require graphing. Here is some review and practice for all possible scenarios. Symbol < > Name Less than Less than or equal to Greater than Greater than or equal to One Variable Graphing circle: open direction: left circle: closed direction: left circle: open direction: right circle: closed direction: right Two Variable Graphing line: dashed shade: below line: solid shade: below line: dashed shade: above line: solid shade: above One-Variable Inequalities Solve as if there is an equal sign in place of the inequality. The inequality sign flips over if you multiply or divide by a negative. If you graph the solution on a number line, the circle used and the direction of the arrow will be based on the inequality sign. Examples: 1. 3x + 4 < 10. 8 x 0 3. 4(x ) 3 7
3x < 6 x 1 4x 8 3 x < x 6 4x 5 x (open circle and left) (closed circle and right) (closed circle and left) 6 Two-Variable Inequalities Solve for y. The inequality sign flips over if you multiply or divide by a negative. If you graph the solution on a coordinate grid, the line used and the direction of the shading will be based on the inequality sign. Solve and graph the inequality. 5. x + y 5 6. x y < 3 7. 8x + 4y 16 y 5-x y < x 3 4y 8x 16 y -x + 5 y > x + 3 y x 4 slope: slope: 1 slope: y-intercept: (0, 5) y-intercept: (0, 3) y-intercept: (0, 4) solid line, shade below dashed line, shade above solid line, shade above O O O 8
Practice Problems 1. 8 x < 6. Solve for x. A. x < -1 B. x > - 1 C. x D. x < 1 E. x > 1. 5x 10 0. Solve for x. F. x 6 G. x 1 H. x 30 J. x > 35 K. the value of x cannot be determined from the information provided. 3. Janet is placing an order for CDs. Each CD costs $3, including tax. Shipping charges are $4 for the total order, and Janet cannot spend more than $5. Which of the following inequalities will help you find c, the number of CDs Janet can order? A. 3c + 7 5 B. 3c - 4 5 C. 3(c + 4) 5 D. 4 (3c) 5 E. 3c + 4 5 F. 3 4. A repairman charges $0 for a service call to a home plus $15 for each hour it takes to repair a broken appliance. What is the maximum amount of time can he spend (in hours) fixing an appliance if the owner of a home cannot spend more than $80? 9
G. 4 H. 5 J. 6 K. 8 Answer Key 1. E. A 3. E 4. G Translating word problems into algebra Review Please see the purplemath.com tutorial for a thorough review of this topic. (The link can be found in your elementary algebra additional resources.) It s fantastic! Since we love the way they explain this concept, we figured there was no need to reinvent the wheel. After you check out that resource, here are some practice problems: Practice Problems 1. The total cost of renting a car is $35 per day for each day the car is rented plus 5 cents for every mile the car is driven. What is the total cost of renting a car for 4 days and driving 300 miles? A. $140 B. $16.50 C. $6 D. $393.50 E. $7,790. It costs a dollars for adult admission to a county fair and c dollars for child admission to the same fair. The difference between the cost of 1 adult ticket and 0 child tickets is $54. Which of the following equations represents this relationship between a and c? 10
F. = 54 G. 10ac = 54 H. 1a 0c = 54 J. 1a + 0c = 54 K. 1c 0a = 54 3. For all real numbers a and c such that the product of c and 3 is a, which of the following expressions represents the sum of c and 3 in terms of a? A. a + 3 B. 3a + 3 C. 3(a + 3) D. E. + 3 4. The contents of a box weigh a certain amount. If you remove fifteen pounds from the box its contents will weigh five pounds more than if you had halved their weight. How much did the box's contents originally weigh? F. 0 G. 5 H. 40 J. 45 K. 50 5. Three monkeys and two giraffes eat a combined 45 kilograms of food in a week. A single giraffe eats two kilograms less than twice what a monkey eats each week. How much does a single monkey eat in a week? A. 1 B. 15 C. 1 D. 7 E. 9 11
6. Delilah Rogers has decided to open a business that produces and sells lawnmowers. In order to begin this venture, she must invest $6 million in a lawn mower production plant. The cost to produce each lawn mower will be $100, and the selling price will be $50. Accounting for the cost of the production plant, which of the following expressions represents the profit, in dollars, that Delilah will realize when x boats are produced and sold? F. 150x 6,000,000 G. 350x 6,000,000 H. 599, 965x J. 50x K. 6,000,000 100x Answer Key 1. B. H 3. E 4. H 5. D 6. F Factoring basic equations Please see the purplemath.com tutorial for a thorough review of this topic. (The link can be found in your elementary algebra additional resources.) It s fantastic! Since we love the way they explain this concept, we figured there was no need to reinvent the wheel. 1