Elementary Algebra
The Elementary Algebra (eighth or ninth grade level) questions test elementary algebraic concepts such as Functions; Polynomial Operations and Factoring Simple Quadratic Expressions; Linear Inequalities with One Variable; and Properties of Integer Exponents and Square Roots.
1. Functions
A function is a set of ordered pairs where no two of the ordered pairs has the same x-value. In a function, each input (x-value) has exactly one output (y-value).
An example of this relationship would be y = x2. Here, y is a function of x, because for any value of x there is exactly one value of y. However, x is not a function of y, because for certain values of y there is more than one value of x. The domain of a function refers to the x-values, while the range of a function refers to the y-values.
If the values in the domain corresponded to more than one value in the range, the relation is not a function. The following is an example of a function question that may appear on the ACT Mathematics Test:
For the function f(x) = x2 − 3x, what is the value of f(5)?
Solve this problem by substituting 5 for x wherever x appears in the function:
f(x) = x2 − 3x
f(5) = (5)2 − (3)(5)
f(5) = 25 − 15
f(5) = 10
2. Polynomial Operations and Factoring Simple Quadratic Expressions
A polynomial is the sum or difference of expressions like 2x2 and 14x. The most common polynomial takes the form of a simple quadratic expression, such as 2x2 + 14x + 8, with the terms in decreasing order.
The standard form of a simple quadratic expression is ax2 + bx + c, where a, b, and c are whole numbers. When the terms include both a number and a variable, such as x, the number is called the coefficient. For example, in the expression 2x, 2 is the coefficient of x.
The ACT Mathematics Test will often require you to evaluate, or solve a polynomial by substituting a given value for the variable, as follows:
For x = −2, 2x2 + 14x + 8 = ?
2(−2)2 + 14(−2) + 8
= 2(4) + (−28) + 8
= 8 − 28 + 8
= −12
You will also be required to add, subtract, multiply, and divide polynomials. To add or subtract polynomials, simply combine like terms and to multiply polynomials, use the distributive property to multiply each term of one polynomial by each term of the other polynomial.
The FOIL Method of multiplication: multiply the First terms, then the Outside terms, then the Inside terms, then the Last terms.
You may also be asked to find the factors or solution sets of certain simple quadratic expressions. A factor or solution set takes the form (x ± some number). Simple quadratic expressions will usually have two of these factors or solution sets. The standard form of a simple quadratic expression is ax2 + bx + c. To factor the equation, find two numbers that when multiplied together will give you c and when added together will give you b.
The ACT Mathematics Test includes questions similar to the following:
What are the solution sets for x2 + 9x + 20?
5 and 4 are two numbers that when multiplied together give you 20, and when added together give you 9. So, (x + 5)(x + 4) are the two solution sets for x2 + 9x + 20
3. Linear Inequalities with One Variable
Linear inequalities with one variable are solved in almost the same manner as linear equations with one variable: by isolating the variable on one side of the inequality. Remember, though, that when multiplying one side of an inequality by a negative number, the direction of the sign must be reversed.
The ACT Mathematics Test will include questions similar to those that follow:
For which values of x is 3x + 4 > 2x + 1?
Follow these steps to solve:
3x + 4 > 2x + 1
3x − 2x > 1 − 4
x > −3
For which values of x is 6x − 32 < 10x + 12?
Follow these steps to solve:
6x − 32 < 10x + 12
6x − 10x < 32 + 12
−4x < 44
Now, since you have to divide both sides by –4, remember to reverse the inequality sign:
x > −11
4. Properties of Integer Exponents
The ACT Mathematics Test will assess your ability to multiply and divide numbers with exponents. The following are the rules for operations involving exponents:
