Elementary Algebra Practice Test for the ACCUPLACER® test
√ 4 × √ 5 =
√ 4 ⋅ √ 5 = √ 20 = √ 2*2*5 = 2 √ 5
4(| − 3 − 2|) − 5 =
Start with the numbers inside the parentheses. 4(| − 3 − 2|) − 5 4(| − 5|) − 5 Remember that the two vertical lines around “- 3 - 2” mean “absolute value.” The absolute value of -5 is 5, so: 4⋅5 − 5 = 15
−2⁄3 + 1⁄6 ⋅ (−2) = ____
Following the “Order of Operations”, first multiply 1⁄6 by −2 to get −2⁄6 so it becomes: −2⁄3 + (−2⁄6) Reduce 2⁄6 to 1⁄ and you have: −2⁄3 + (−1⁄3) = −1
Which of the following sequence of numbers lists the numbers from the least to the greatest?
There are several ways to approach a test item like this one. If the same numbers are given in each answer choice, simply convert them all to the same format. In this case, fractions and mixed numbers with the denominator of 6 would work well. Also note the simplification of the absolute value amount to −2⁄3, before changing to −4⁄6. (An absolute value is just the positive value of any number, so the negative of that would be negative.) 7⁄2 = 3 3⁄6 -5⁄6 = -5⁄6 −∣−2⁄3∣ = −4⁄6 1⁄2 = 4⁄8 = 3⁄6 Now, list them in the correct order and compare your listing with the answer choices to find the correct one. −5⁄6 < −4⁄6 < 3⁄6 < 3 3⁄6 For a test item in which the numbers in each of the answer choices are not the same, try to find errors that immediately “stick out” to you. For instance, in the choice −5⁄6 <1⁄2 < −∣−2⁄3∣ < 4⁄8 < 7⁄2, you’ll immediately notice that 1⁄2 is listed as less than 48 and you know those are equal, so that choice is wrong. There are pretty obvious inaccuracies in the other two incorrect choices for this problem, as well. So, you’d be able to narrow it down to only one possible choice, which you would need to check with the first process, above, to be sure.
4(−7+5) − (−2)(6−11) = ____
4(−7+5) − (−2)(6−11) In a problem such as this one, it is necessary to follow the order of operations, PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. Perform these steps, from left to right, in this order. So, performing the operations inside parentheses first: 4(−7+5) − (−2)(6−11) 4⋅(−2) − (−2)⋅(−5) Then, since there are no exponents, do multiplication next. −8 − 10 −18 PEMDAS can be remembered with the pneumonic, “Please Excuse My Dear Aunt Sally.”
Distribute x by multiplying with the two terms inside the parentheses: x(x2−3)=x3−3x. Then, combine with the other terms: x3−3x−2x2+5. Finally, arrange terms in such a way that the exponents of the variable x are in descending order with the constant as the last term. Thus: x3−2x2−3x+5.
Write the expression out as (5x+2y)(5x+2y). Then, multiply the expressions together by FOILing (multiple the First terms, the Outer terms, the Inner terms, and the Last terms). First terms: (5x)⋅(5x) = 25x2 Outer terms: (5x)⋅(2y)=10xy Inner terms: (2y)⋅(5x)=10xy Last terms: (2y)⋅(2y)=4y2 Combining all terms: 25x2+10xy+10xy+4y2 Add similar terms: 25x2+20xy+4y2
(x2+ 8x + 15)⁄(x + 3) =
Factor the numerator using the reverse FOIL method: x2+8x+15=(x+5)(x+3). The (x+3) term on both the numerator and the denominator cancels out and we are left with (x+5), the answer.
Rearrange the equation so only the term with the x (in this equation, the ‘xy’ term) is on the left side. Thus: xy= 2y2−5y+4 Then, divide both sides by y to get an expression for x. The equation becomes: x=2y−5+4⁄y = 2y+4⁄y
Evaluate the following expression for x = 6 and y = -1: 3y2−4x+2xy
Plug in the given values for x and y. Substituting: 3y2−4x+2xy 3(−1)2−4(6)+2(6)(−1) Perform the operations, from left to right: 3−24+(−12) −33
Solve the equation for x, given y = 3 9x−12+y2 = 5
Plug in the given value y. Then rearrange terms so that only x is on the left side. 9x−12+y2=5 9x−12+32=5 9x−12+9=5 9x=12−9+5=8 x= 8⁄9
What are the roots of the polynomial (x+1)(x2−x−6) = 0?
Find the roots of the polynomial by setting each expression equal to 0 and solving for x. For the expression (x+1) = 0: x+1 = 0 x=−1 Next, do the expression (x2−x−6)=0 x2−x−6 = 0 Factoring out: (x+2)(x−3) = 0 x = −2 and x = 3. Therefore, the roots are -1, -2 and 3
(-3)(2+6)⁄4 - 2 =
Start with the numbers inside the parentheses and then proceed with multiplication and division. (-3)(2+6)⁄4-2 (-3)(8)⁄4-2 = -24⁄4-2 = -6-2 = -8
Which of the following is equal to (x+3)(x2+3x−5)?
To solve (x+3)(x2+3x−5), multiply x by all of the terms in the second expression, and then multiply 3 by all of the terms in the second expression. Then, add to combine like terms, arranging the exponents in descending order. (x+3)(x2+3x−5)=(x3+3x2−5x)+(3x2+9x−15)=x3+6x2+4x−15
x = 3 and x = -3 are both solutions to which of the following equations?
Only the equation x2−9 = 0 holds true for the values of x=3 and x=-3. (3)2−9=9−9=0 and (−3)2−9=9−9=0
For x > 0, 3y⁄x - y⁄2x + 2y⁄4x =
Add the different terms (which are all in fraction form) by making sure they have the same denominator. Multiply the first term by 4⁄4 and the second term by 2⁄2 so all 3 terms have 4x as denominator. 3Y⁄x - y⁄2X + 2y⁄4x 12y⁄4x - 2y⁄4x + 2y⁄4x (12y-2y+2y)⁄4x) = 12y⁄4x Simplifying, 12y⁄4x = 3y⁄x
Which of the following correctly lists the numbers in order from greatest to least?ub>2
5⁄2 must be the greatest because it is the only given fraction that is greater than one. Next must be 3⁄4 because it is the only remaining positive fraction. Finally, −1⁄4 is closer to 0 than −1⁄2 is, so −1⁄4 is greater than −1⁄2.
Which of the following lists the fractions in order from greatest to least when x = -1?
Plug in x = -1 into all terms. This gives us the numbers: -3⁄(2)(-1) = 3⁄2 1⁄(2)(-1) = -1⁄2 2(-1) = -2 From the greatest to least, the order would be: 3⁄2, 2⁄3, -1⁄2, -2 or -3⁄2x > 2⁄3 > 1⁄2x > 2x
3 - 7x⁄2 < 10
Our goal is to isolate the x. 3 - 7x⁄2 < 10 3 - 3 - 7x⁄2 < 10 - 3 (-2⁄7 ⋅ − 7x⁄2 > 7 ⋅ (-2⁄7) x>−2. Flip the inequality sign because we divided by a negative number.
Solve the following system of equations for (x,y), 3x+4y = 25, x−2y = 5
Start with the second equation and isolate x: x−2y=5 x=5+2y Substitute this result to x in the first equation: 3(5+2y)+4y = 25 15+6y+4y = 25 10y = 25−15 10y = 10 y = 10⁄10 = 1 Then, plug y = 1 into either equation (preferably the simpler one): x = 5+2(1) = 7
Which of the following is a factor of x2−x−12 ?
To factor the given polynomial: x2−x−12 Think of 2 numbers which will result to -12 when multiplied, and will result to -1 when added. (x+3)(x−4). Take note that: 3x-4 = -12, and 3+(-4) = -1. Either of (x+3) and (x−4) are factors, but only (x+3) is given in the choices; hence, it is the correct answer.
John is shopping for supplies for his office. Let x equal the number of pens he buys and y equal the number of pencils. If pens cost $0.45 and pencils cost $0.15, which of the following represents the total cost of his purchase?
If pens cost $0.45 and pencils cost $0.15, the total cost of his purchase is represented by the equation: TotalPurchase=$0.45x+$0.15y This is not given as an option. However, it should be noticed that both 0.45 and 0.15 are divisible by 3, meaning a 3 can be factored out of both terms: 3($0.15x+$0.05y) which when expanded gives: $0.45x+$0.15y. Therefore, it is the correct answer.
Find the roots of the following polynomial: x2 + 2x − 15 = 0
The roots of a polynomial equation are the values that when substituted into the equation yield a true statement. ‘Finding roots’ is another way of describing, ‘solve for x.’ Note: The roots are not the same as the factors. The factors are the values that can be multiplied together to equal the original equation. You will find the factors first, then set each of them equal to 0 to find the roots. Finding the factors: x2+2x−15 = 0 Think of 2 numbers that result in -15 when multiplied, and result in 2 when added. That gives us the factors: (x−3)(x+5) = 0 Take note that: −3⋅5 = −15, and −3+5 = 2. Now, find the roots. Equate each factor to 0: x−3 = 0 x = 3 x+5 = 0 x = −5 The roots of the expression are 3 and -5.
7 and -4 are the roots of which of the following polynomials?
To get the polynomial with 7 and -4 as roots, Use FOIL method and multiply the factors: (x−7)(x+4) = x2−3x−28 Hence, the polynomial x2−3x−28 has the roots 7 and -4.
Which of the following are factors of x2−16 = 0?
This is a problem on factoring a difference of squares because there is an x2 term, a subtraction sign, no x term, then a constant. Start with: (x+_)(x−_) The two numbers will be the same but opposite in sign, and will be the square root of 16. Hence: (x+4)(x−4) These are the factors of the given expression.
Stacy has a mixture of quarters and dimes in her pocket. The total value of the coins is $2.25. If she has a total of 12 coins, how many of each coin does she have?
To solve this word problem, set up a system of equations with the given information. Let q = the number of quarters and d = the number of dimes. She has 12 coins total, so we know that: q+d = 12 (equation 1) q = 12−d We also know the total value of the coins is $2.25, so we have: (equation 2) 0.25q+0.1d = 2.25 Plug in q from equation 1 into equation 2: 0.25(12−d)+0.1d = 2.25 3−0.25d+0.1d = 2.25 −0.15d = 2.25−3 d = −0.75−0.15 = 5 12−5 = 7 Therefore, there are 5 dimes and 7 quarters.
The sum of 2 numbers is 27. Three times the first number is equal to 3 less than the second number. What is the value of the larger number?
et up a system of equations with the given information. Let the 2 numbers be represented by x and y. Write “the sum of the two numbers is 27” as: x+y = 27 x = 27−y (equation 1) Write “3 times the first number is equal to 3 less than the second number” as: 3x = y−3 (equation 2) Solve equation 2 by using the value of x from equation 1: 3(27−y) = y−3 81−3y = y−3 −4y = −84 4y = 84 y = 21 x = 27−21 = 6
Write the following polynomial as a product of its factors: x2+3x−40
To factor the given polynomial x2+3x−40 Think of 2 numbers that result in -40 when multiplied, and result in 3 when added. (x−5)(x+8)
Steven rents a car for his vacation. The rental agency charges him $78 per day for the car. Furthermore, if he drives more than 500 miles, there will be an additional charge of 20 cents per mile for each mile over 500. If Steven rents the car for 5 days and drives a total of 563 miles, how much does he owe the rental agency?
The total rental cost for the car is the rental for 5 days + the rental in excess of 500 miles Total rental cost = $78(5)+$0.20(563−500) = $390+$12.60 = $402.60
Which of the following are possible values for x if 2x=(x2)
Try the given choices by plugging the values into the given equation. We can check each: For x = 0: 20 ≠ 02 For x = 1: 21 ≠ 12 For x = 2: 22 = 22 For x = 3: 23 ≠ 32 For x = 4: 24 = 42 For x = 5: 25 ≠ 52 So, “2 and 4” is the correct solution because all the other solutions contain at least one wrong answer.
√ 12 √ 3 is equal to all but which of the following?
√ 12 √ 3 can be simplified as √ 2*2*3 √ 3 = 2 √ 3 √ 3 = 2(3) = 6 Each of the following are equal to 6: √ 36 = 6 √ 9 √ 4 = 3*2 = 6 1⁄2 √ 144 = 1⁄2 * 12 = 6 Only 3 √ 9 = 3*3 = 9, hence, not equal to 6 or √ 12 √ 3
John’s business started with 25 people. Since then, they have brought on 10 additional employees every year. Which of the following equations gives the number of employees in John’s company (y) as a function of the number of years (x)?
The problem says that y is the number of employees in the company. It starts with 25 people. At the start: y = 25 For every year (x), 10 employees are added: y = 25+10x Write the expression in the normal format, that is, terms with variable first, then the constant: y = 10x+25
A rectangular pool is 50 feet wide. The pool is 50% longer than it is wide, and 1⁄ as deep as it is wide. What is the volume of the pool?
The formula for volume is Volume = length⋅width⋅depth. W = 50feet L = 50+50%(50) =50+25 = 75feet D = 15(50) = 10feet Volume = LWD = (50)(75)(10) = 37,500feet3
What is the product of (x+2)(x2−3x+5)?
Just like when multiplying two binomials, each term in the first set of parentheses must be multiplied by all of the terms in the second set. (x+2)(x2−3x+5)=x(x2−3x+5)+2(x2−3x+5)=(x3−3x2+5x)+(2x2−6x+10)=(x3−x2−x+10)
If the circumference of a circle is 10π, then what is its area?
The formula for the circumference of a circle is: C=2πr. It is given in this question that: C=10π Equating C=2πr=10π r=10π2π=5 The formula for the area of a circle is A=πr2. Plug in r=5: A=π(52)=25π.
A triangle has a height of 7 inches and an area of 35 inches2. What is the length of half of the base of the triangle?
The formula for the area of a triangle is Area = 1⁄2 ⋅ Base ⋅ Height. Plug in the given values: 35in2 = 1⁄2 ⋅ Base ⋅ 7in Base = (35)(2)⁄7 = 10 The question asks for half of the length of the base, so the correct answer is 5inches.
Emily’s Shake Shack sells two types of milkshakes: chocolate and vanilla. If a chocolate shake costs $2.75 and a vanilla shake costs $2.50, which of the following equations would accurately give Emily’s revenue (R) based on the number of chocolate (C) and vanilla (V) shakes she sold?
Emily’s total revenue (R) equals total chocolate shakes sold (C) at $2.75 plus total vanilla shakes sold (V) at $2.50. Hence: R=$2.75C+$2.50V