TEAS® Math Study Guide
Algebra
Algebraic Expressions
An algebraic expression is a mathematical phrase made up of:
- Numbers (constants)
- Variables (like \(x\) or \(y\)
- Operations (addition, subtraction, multiplication, division)
Examples: \(3y, 2x + 5, a^2 + 6a – 9\)
Tip: You cannot solve an expression — you can only simplify or evaluate it.
Simplifying Expressions
When not given the value of a variable, you can simplify an expression, where you combine groups of terms into their simplest form. To simplify, combine like terms within an expression. Like terms are those with the same variables and same powers (e.g. \(3x\) and \(x\); \(13\) and \(435\); \(y^3\) and \(7y^3\)). You should always simplify using the order of operations (PEMDAS: Parentheses → Exponents → Multiplication/Division → Addition/Subtraction).
Example:
Simplify the expression:
\((2x^2- 5)+ (x^2 – 2x + 4 – x)\)
First, per the order of operations, see if anything can be simplified within the parentheses.
In the second parenthetical, \(– 2x\) and \(– x\) can be combined to create \(– 3x\), producing the new equation:
\((2x^2- 5) + (x^2 – 3x + 4)\)
Now we can combine like terms across both sets of parentheses.
\(– 2x^2\) and \(x^2\) can be added (since the sign between the parentheticals is an addition sign) to yield \(– 3x^2\).
\(– 5\) and \(+ 4\) can be added to yield \(-1\).
\(– 3x\) does not have a like term, so it will remain the same.
The simplified expression is:
\(3x^2 – 3x – 1\)
Evaluating Expressions
To evaluate an algebraic expression, follow these steps:
- Identify the value of the variable.
- Substitute the variable with the given value.
- Simplify using the correct order of operations.
Example: Evaluate \(2x^2 – x + 12\) when \(x = 3\).
Step 1: Substitute:
\(=2(3)^2 – 3 + 1\)
Step 2: Exponents first, then multiplication:
\( = 2(9) – 3 + 1 = 18 – 3 + 1\)
Step 3: Simplify:
\(16\)
Tip: Always use parentheses when substituting a negative number.
One-Variable Equations
An equation is a mathematical statement that two expressions are equal. We solve it by finding the value of the variable that makes the equation true.
Example: Solve: \(3x + 2 = 11\)
Step 1: Subtract 2 from both sides: \(3x = 9\)
Step 2: Divide both sides by 3: \(x = 3\)
Forming equations for solving word problems involves translating words into math.
Example: Sam bought 4 pencils and a notebook for $10. Each pencil costs $1. How much was the notebook?
Let \(x\) be the cost of a notebook and \(y\) be the cost of a pencil, and form the equation:
\(4y + x = 10\)
The value of \(y\) is given as 1. Substitute it in the equation.
\(4(1) + x = 10\)\(4 + x = 10\)
\(x = 10 – 4 = 6\)
Thus, one notebook costs $6.
Inequalities
An inequality compares two expressions using one of the following signs:
- \(\lt\) (less than)
- \(\le\) (less than or equal to)
- \(\gt\) (greater than)
- \(\ge\) (greater than or equal to)
Inequalities can be solved using the same steps as equations, with the addition of a special rule.
Example: Solve: \(2x − 4 \lt 10\)
Step 1: Add 4 to both sides:
\(2x \lt 14\)
Step 2: Divide both sides by 2:
\(x \lt 7\)
Thus, all values of \(x\) less than \(7\) are solutions.
Tip: When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
Real-World Problems
Everyday problems involving budgeting, calculating distances, counting product quantities, and comparing prices can be solved by setting up equations and inequalities.
Example: You have $100. You want to buy books that cost $12 each. What’s the greatest number of books you can buy?
Let \(x\) be the number of books:
\(12x \le 100 ⇒ x \le 8.33 \)
At most, you can buy 8 books (since you can’t buy part of a book).
Tips:
| Concept | What You Do |
| Expression | Simplify or evaluate |
| Equation | Solve for the variable |
| Inequality | Solve |
| Real-world word problems | Translate into algebra and solve |