TEAS Math: Algebra Basics — Solving Equations and Expressions

Algebra intimidates a lot of TEAS test-takers, but it is really just arithmetic with a placeholder. Once you understand what a variable represents and learn a few reliable rules for keeping equations balanced, most TEAS algebra questions become predictable—and quick. This guide builds from the ground up: variables and expressions, the order of operations, solving one- and two-step equations, working with inequalities, and translating word problems into algebra.

The TEAS Math section includes algebra under its 'Algebraic Applications' content. You will not be asked to factor complex polynomials, but you will need to solve for an unknown, simplify expressions, and set up equations from real-world descriptions—often in a healthcare or everyday context.

Variables, Terms, and Expressions

A variable is a letter (like x or n) that stands in for an unknown number. The pieces of an expression are called terms, and terms separated by + or − signs can sometimes be combined.

• Variable: a symbol for an unknown value, e.g. the x in 3x + 4.
• Coefficient: the number multiplied by a variable; in 3x, the coefficient is 3.
• Constant: a number on its own with no variable, e.g. the 4 in 3x + 4.
• Term: a single part of an expression separated by + or −; 3x and 4 are both terms.
• Like terms: terms with the exact same variable part, e.g. 3x and 5x (these can be combined to 8x), but 3x and 3x² cannot.

The Order of Operations (PEMDAS)

Before solving equations, you must evaluate expressions in the correct order. Use PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

• Parentheses first: simplify anything inside grouping symbols.
• Exponents next: evaluate powers and roots.
• Multiplication and Division: work left to right—whichever comes first.
• Addition and Subtraction: work left to right—whichever comes first.

Example: Evaluate 5 + 3 × (8 − 2)² ÷ 6. Parentheses: (8 − 2) = 6. Exponent: 6² = 36. Now 5 + 3 × 36 ÷ 6. Multiplication/division left to right: 3 × 36 = 108, then 108 ÷ 6 = 18. Finally 5 + 18 = 23.

The Golden Rule of Solving Equations

To solve an equation means to find the value of the variable that makes both sides equal. The single most important rule: whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced—think of it as a scale.

Your goal is to ISOLATE the variable (get it alone on one side) by undoing operations with their opposites: addition undoes subtraction, multiplication undoes division, and vice versa.

One-Step Equations

Solve x + 7 = 12. The variable has 7 added to it, so subtract 7 from both sides: x + 7 − 7 = 12 − 7, giving x = 5. Check: 5 + 7 = 12. Correct.

Solve 4x = 20. The variable is multiplied by 4, so divide both sides by 4: 4x ÷ 4 = 20 ÷ 4, giving x = 5. Check: 4 × 5 = 20. Correct.

Two-Step Equations

Two-step equations require undoing operations in reverse PEMDAS order: deal with addition/subtraction first, then multiplication/division.

Solve 3x + 5 = 20. Step 1: subtract 5 from both sides → 3x = 15. Step 2: divide both sides by 3 → x = 5. Check: 3(5) + 5 = 15 + 5 = 20. Correct.

Solve (x ÷ 2) − 4 = 6. Step 1: add 4 to both sides → x ÷ 2 = 10. Step 2: multiply both sides by 2 → x = 20. Check: 20 ÷ 2 − 4 = 10 − 4 = 6. Correct.

Distributing and Combining Like Terms

When an equation has parentheses, use the distributive property first: a(b + c) = ab + ac. Then combine like terms before isolating the variable.

Solve 2(x + 3) = 16. Distribute: 2x + 6 = 16. Subtract 6: 2x = 10. Divide by 2: x = 5. Check: 2(5 + 3) = 2(8) = 16. Correct.

Solve 5x − 2x + 4 = 19. Combine like terms (5x − 2x = 3x): 3x + 4 = 19. Subtract 4: 3x = 15. Divide by 3: x = 5.

Solving Inequalities

Inequalities use the symbols < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). You solve them exactly like equations, with ONE crucial exception.

Solve 3x + 2 ≤ 14. Subtract 2: 3x ≤ 12. Divide by 3 (positive, no flip): x ≤ 4. The solution is all values of x that are 4 or less.

Translating Word Problems into Equations

Many TEAS algebra questions are written in words. The skill is translating English phrases into math symbols. Learn these keyword cues:

• 'Sum,' 'more than,' 'increased by,' 'total' → addition (+).
• 'Difference,' 'less than,' 'decreased by,' 'fewer' → subtraction (−).
• 'Product,' 'times,' 'of,' 'twice' → multiplication (×).
• 'Quotient,' 'per,' 'divided by,' 'ratio' → division (÷).
• 'Is,' 'equals,' 'results in,' 'will be' → equals (=).

Example: 'A nurse works 5 more hours this week than last week. If she worked 42 hours this week, how many did she work last week?' Let x = last week's hours. Then x + 5 = 42, so x = 37 hours.

Common Algebra Mistakes to Avoid

• Forgetting to apply an operation to BOTH sides of the equation.
• Ignoring the order of operations and adding before multiplying.
• Combining unlike terms (e.g., adding 3x and 5 to get 8x).
• Forgetting to flip the inequality sign when dividing by a negative.
• Misreading 'less than' and reversing the subtraction order.
• Skipping the check step and submitting a sign-error answer.

Your Algebra Study Plan

Algebra rewards repetition. Practice 10–15 mixed equation problems per day, always checking your answers. Start with one-step problems until they feel automatic, then move to two-step and distributive problems, and finish with word-problem translation. Keep our formula sheet nearby as you work.

Master these fundamentals and you will be able to handle the bulk of TEAS algebra questions with speed and confidence. The patterns repeat—once you recognize them, solving for x becomes one of the easier parts of the Math section.