3.5 Algebra and Equations

Why Algebra Is on the TEAS

The ATI TEAS 7 Mathematics section contains 38 questions (34 scored, 4 unscored pretest items) answered in 57 minutes, and they fall into two reporting areas: Numbers and Algebra and Measurement and Data. Algebra lives in the first area, where you are asked to solve equations, simplify expressions, and turn sentences into math.

The test rewards a calculator-light, structured approach: an on-screen four-function calculator is provided, but the thinking must be yours. In nursing practice the same algebra solves for an unknown dose, an unknown drip rate, or an unknown concentration, so the exam is really checking whether you can find a missing value reliably.

Expressions vs. Equations

An expression is a combination of numbers, variables, and operations with no equals sign (for example, 3x + 5). An equation sets two expressions equal (for example, 3x + 5 = 20) and can be solved for the variable. Knowing the vocabulary keeps the question types straight.

Combining Like Terms

Like terms share the same variable raised to the same power, so 3x and 5x are like terms but 3x and 3x² are not. Add or subtract only the coefficients; the variable part stays the same.

Example: Simplify 5x² + 3x - 2x² + 7x - 4. Group the x² terms: 5x² - 2x² = 3x². Group the x terms: 3x + 7x = 10x. The constant -4 stands alone. Answer: 3x² + 10x - 4.

Solving One-Step Equations

The goal is always to isolate the variable by doing the inverse (opposite) operation to both sides, which keeps the equation balanced. Addition undoes subtraction, multiplication undoes division, and so on.

Solving Two-Step and Multi-Step Equations

When two operations stand between you and the variable, undo them in reverse order of operations: first remove anything added or subtracted, then remove anything multiplied or divided. If like terms or parentheses appear, simplify those first.

Worked Example (two-step): Solve 3x + 7 = 22.

1. Subtract 7 from both sides: 3x = 15.
2. Divide both sides by 3: x = 5.
3. Check: 3(5) + 7 = 15 + 7 = 22 ✓

Worked Example (variable on both sides): Solve 5x - 4 = 2x + 11.

1. Subtract 2x from both sides to collect variables on the left: 3x - 4 = 11.
2. Add 4 to both sides: 3x = 15.
3. Divide by 3: x = 5. Check: 5(5) - 4 = 21 and 2(5) + 11 = 21 ✓

Translating Word Problems

Many TEAS algebra items are sentences you must convert to symbols. Watch the order-sensitive phrases especially closely.

• Sum / more than / increased by → addition (+)
• Difference / less than / decreased by → subtraction (-), and less than flips the order
• Product / times / of → multiplication (×)
• Quotient / per / ratio → division (÷)
• Is / equals / results in / gives → the equals sign (=)

Worked Example (translation): "Three less than four times a number equals 17." "Four times a number" is 4x. "Three less than" that means 4x - 3 (NOT 3 - 4x). Setting it equal to 17 gives 4x - 3 = 17. Solving: add 3 → 4x = 20 → x = 5.

Real-World Example (nursing setup): An order reads "give 0.5 mg per kilogram." For a patient who weighs w kilograms the dose is the expression 0.5w milligrams. If the prescribed total dose is 35 mg, the equation 0.5w = 35 solves to w = 70 kg — the same one-step division you practiced above.

Solving Inequalities

An inequality uses <, >, ≤, or ≥ instead of = and is solved with the same inverse-operation steps — with one critical twist. When you multiply or divide both sides by a negative number, reverse (flip) the inequality sign. For example, -2x < 8 becomes x > -4 after dividing by -2. Forgetting to flip is the single most common inequality mistake the TEAS tries to catch.

Recap

Treat every equation as a balance scale: whatever you do to one side, do to the other, undoing operations in reverse order until the variable stands alone. Combine like terms first, translate word phrases literally (minding "less than" and "per"), flip the sign when an inequality is scaled by a negative, and always substitute your answer back to confirm both sides match.