3.1 Simplifying Algebraic Expressions

Overview of Algebraic Expressions on the TEAS 7

Algebraic expressions form the core foundational language of mathematical reasoning tested on the ATI TEAS 7 Mathematics section. An algebraic expression is a mathematical phrase containing numbers, variables (letters representing unknown quantities), and operation symbols (such as addition, subtraction, multiplication, and division). Unlike algebraic equations, expressions do not contain an equals sign (=) and therefore cannot be solved for a single numerical value unless specific values are assigned to the variables.

In healthcare and nursing practice, algebraic expressions are used daily to model physiological processes, convert patient parameters, and compute medication administration rates. For example, calculating a patient's Body Surface Area (BSA), estimating pediatric maintenance fluid requirements, or adjusting insulin sliding scales relies on evaluating multi-variable algebraic expressions. Mastering expression simplification ensures you can manipulate these clinical formulas quickly and without calculation errors under exam conditions.

---

The Order of Operations (PEMDAS / GEMDAS)

When simplifying or evaluating complex algebraic expressions, you must strictly follow the standard mathematical hierarchy known as the Order of Operations, commonly remembered by the acronym PEMDAS (or GEMDAS, where G stands for Grouping symbols):

1. P / G — Parentheses & Grouping Symbols: Evaluate all operations inside parentheses (), brackets [], braces {}, absolute value bars ||, and above or below fraction bars.
2. E — Exponents: Evaluate all exponential expressions and roots (e.g., $3^2 = 9$, $\sqrt{16} = 4$).
3. MD — Multiplication and Division: Perform all multiplication and division in order from left to right as they appear in the expression. Multiplication does not take precedence over division; they share equal priority.
4. AS — Addition and Subtraction: Perform all addition and subtraction in order from left to right as they appear. Addition does not take precedence over subtraction; they share equal priority.

Exam Warning — Left-to-Right Priority: A very frequent TEAS trap involves expressions with consecutive division and multiplication, such as $24 \div 4 \times 2$. Students who mistakenly believe 'M comes before D' calculate $4 \times 2 = 8$ first, giving $24 \div 8 = 3$ (INCORRECT). The correct procedure executes left to right: $24 \div 4 = 6$, then $6 \times 2 = 12$.

---

The Distributive Property and Signed Numbers

The distributive property allows you to remove grouping symbols by multiplying the factor outside the parentheses by every term inside the parentheses:

$a(b + c) = ab + ac$ $a(b - c) = ab - ac$

Distributing Negative Signs

When distributing a negative number or a negative sign, every term inside the parentheses flips its sign:

$-a(b + c) = -ab - ac$ $-a(b - c) = -ab + ac$

Step-by-Step Worked Example 1: Distributive Simplification

Simplify the expression: $4(3x - 5) - 2(2x - 7)$

• Step 1 (Distribute the 4): $4 \cdot 3x - 4 \cdot 5 = 12x - 20$
• Step 2 (Distribute the -2 across both terms): $-2 \cdot 2x - (-2) \cdot 7 = -4x + 14$
• Step 3 (Combine the distributed parts): $12x - 20 - 4x + 14$
• Step 4 (Group like terms): $(12x - 4x) + (-20 + 14)$
• Step 5 (Simplify coefficients): $8x - 6$

---

Identifying and Combining Like Terms

An algebraic term consists of a coefficient (numerical multiplier) and a variable part (variables raised to exponents). Like terms are terms that contain the exact same variables raised to the exact same powers. Only the coefficients of like terms can be added or subtracted.

Step-by-Step Worked Example 2: Multi-Variable Simplification

Simplify the expression: $7x^2 + 4xy - 3x + 2x^2 - 9xy + 8x - 5$

• Step 1 (Identify and group like terms by variable powers):
  • $x^2$ terms: $7x^2 + 2x^2 = 9x^2$
  • $xy$ terms: $4xy - 9xy = -5xy$
  • $x$ terms: $-3x + 8x = +5x$
  • Constant terms: $-5$
• Step 2 (Write in standard polynomial form): $9x^2 - 5xy + 5x - 5$

---

Evaluating Algebraic Expressions

To evaluate an algebraic expression, replace each variable with its given numerical value and follow PEMDAS to calculate the final numerical result. Always wrap substituted values in parentheses to avoid sign errors.

Step-by-Step Worked Example 3: Evaluating with Negative Inputs

Evaluate the expression $2a^2 - 3ab + 4b^2$ when $a = -3$ and $b = 4$.

• Step 1 (Substitute values with parentheses): $2(-3)^2 - 3(-3)(4) + 4(4)^2$
• Step 2 (Evaluate exponents first):
  • $(-3)^2 = +9$ (Note: negative times negative yields positive)
  • $(4)^2 = 16$
  • Expression becomes: $2(9) - 3(-3)(4) + 4(16)$
• Step 3 (Perform multiplication from left to right):
  • $2 \cdot 9 = 18$
  • $-3 \cdot (-3) \cdot 4 = +36$
  • $4 \cdot 16 = 64$
• Step 4 (Perform addition): $18 + 36 + 64 = 118$

---

Healthcare Application: Pediatric Dosing Formula

In clinical nursing, dosage formulas often require expression evaluation. Consider Clark's Rule for pediatric dosing adjustments based on body weight:

$\text{Child Dose} = \frac{\text{Child Weight in lbs}}{150} \times \text{Adult Dose}$

Suppose a clinical trial uses an adjusted maintenance dose expression: $D = 12W - 2.5(A + 4)$, where $W$ is patient weight in kg and $A$ is age in years.

• Clinical Problem: Evaluate the daily maintenance dose $D$ for a 6-year-old child weighing $22\text{ kg}$.
• Step 1 (Substitute $W = 22$ and $A = 6$): $D = 12(22) - 2.5(6 + 4)$
• Step 2 (Parentheses first): $6 + 4 = 10 \implies D = 12(22) - 2.5(10)$
• Step 3 (Multiply): $12 \cdot 22 = 264$ and $2.5 \cdot 10 = 25$
• Step 4 (Subtract): $D = 264 - 25 = 239\text{ mg/day}$

---

TEAS 4-Function Calculator Strategy

Crucial Calculator Warning: The built-in calculator on the TEAS 7 exam is a basic 4-function calculator. It does NOT possess order-of-operations logic, memory parentheses, or scientific algebraic entry. If you enter 3 + 4 * 5 =, the calculator will perform (3 + 4) * 5 and output 35, which is WRONG (the correct PEMDAS value is 23).

1. Work Out Operations on Scratch Paper: Always identify multiplication, division, and exponent blocks on scratch paper first.
2. Use the Calculator for Individual Calculations Only: Use the calculator to perform $12 \times 22 = 264$, write $264$ on your scratch pad, then calculate $2.5 \times 10 = 25$, write $25$ down, and finally subtract $264 - 25 = 239$.
3. Never Chain Mixed Operations: Do not enter addition and multiplication sequentially into the on-screen display without pressing equals and clearing between steps.

---

Common Student Traps & How to Avoid Them

1. The Distributed Negative Sign Error: Writing $-(3x - 5)$ as $-3x - 5$. Remedy: Always draw distribution arrows on scratch paper and remember that a negative outside flips every inside sign: $-3x + 5$.
2. Combining Unlike Exponents: Adding $4x^3 + 2x^2$ to get $6x^5$. Remedy: Remind yourself that exponents never change during addition/subtraction. You can only add/subtract coefficients of identical variable power terms.
3. Squaring Negative Numbers without Parentheses: Thinking $-4^2 = 16$. Remedy: Remember $-4^2$ means $-(4 \times 4) = -16$. Only $(-4)^2$ equals $+16$.