Equations and Inequalities

To manipulate math, we must first agree on what we are looking at. An equation is a mathematical statement asserting equality between two expressions. It is a perfectly balanced scale. When we say two things are equal, we are stating a fact about the universe in that moment.

For the ATI TEAS 7, you will primarily deal with a one-variable linear equation. This is an equation that has a single unknown quantity and no exponents greater than one on the variable. Within these equations, you will encounter two types of numbers:

• A mathematical constant is a fixed numerical value that stands alone without a variable in an algebraic expression. Think of this as the baseline 5 mL flush you always use, regardless of the patient's weight.
• A coefficient is a numerical value that is multiplied by a variable in an algebraic term. If you administer 2 mg of medication per kilogram of a patient's weight, the "2" is the coefficient of the variable weight ($x$), written as $2x$.

The primary goal of solving an algebraic equation is to isolate the variable on one side of the equal sign. You want to peel away the layers of numbers until the unknown stands entirely alone.

Important Note: Variables can be validly isolated on either the left or the right side of the equal sign during the equation-solving process. $x = 5$ is mathematically identical to $5 = x$. Do not let a right-sided variable confuse you.

How do we isolate the variable without destroying the delicate balance of our equation? By utilizing inverse operations—pairs of mathematical operations that undo the effects of each other.

• Addition and subtraction are inverse operations.
• Multiplication and division are inverse operations.

If a variable has $4$ added to it, we subtract $4$ to undo it. If it is multiplied by $3$, we divide by $3$. However, whatever surgical adjustment we make to one side of our scale, we must exactly mirror on the other side. This is guaranteed by the properties of equality:

• The addition property of equality states that adding the same number to both sides of an equation does not change the truth value of the equation.
• The subtraction property of equality states that subtracting the same number from both sides of an equation maintains equality.
• The multiplication property of equality states that multiplying both sides of an equation by the same non-zero number maintains equality.
• The division property of equality states that dividing both sides of an equation by the same non-zero number maintains equality.

Real-world equations rarely arrive neatly pre-packaged. Before you start shifting weights across the equal sign using inverse operations, you must organize what you have.

Combining Like Terms

First, look for like terms. These are terms in an algebraic expression whose variables and exponent values are exactly the same (e.g., $3x$ and $5x$). Like terms can be combined into a single term by adding or subtracting their numerical coefficients. Just as you wouldn't chart heart rate and respiratory rate as a single combined number, you cannot add $3x$ and $4y$. But $3x + 5x$ easily combines to $8x$.

The Distributive Property

Often, terms are locked inside parentheses, representing a grouped quantity. The distributive property is used to remove parentheses when solving equations by multiplying the term outside the parentheses by each term inside.

The distributive property is algebraically defined as $a(b + c) = ab + ac$.

If a patient needs 3 doses of a cocktail containing 5 mg of Drug A and $x$ mg of Drug B, represented as $3(5 + x)$, the total requirement is $15 + 3x$.

Clearing Fractions

Fractions can obscure the simplicity of an equation. A linear equation containing fractions can be simplified by multiplying every term on both sides by the least common denominator (LCD) of the fractions. This clever use of the multiplication property of equality wipes the fractions out completely, leaving you with whole, manageable numbers.

Once you isolate your variable, you will arrive at one of three possible realities.

1. Exactly One Solution: A one-variable linear equation with exactly one solution occurs when solving yields a single distinct value for the variable (e.g., $x = 12$). This is the standard outcome: the patient needs exactly 12 mg of the drug.
2. No Solution: Sometimes, the math breaks down. A one-variable linear equation has no solution if simplifying both sides algebraically results in a false statement like $3 = 5$. This indicates an impossible scenario—like programming an IV pump to deliver two different volumes simultaneously.
3. Infinitely Many Solutions: Conversely, an equation has infinitely many solutions if simplifying both sides algebraically results in a universally true statement like $4x = 4x$. This means that no matter what value you choose for $x$, the equation holds true.

Regardless of your result, a careful practitioner always verifies their work. Checking a solution involves substituting the calculated variable value back into the original equation to verify that it produces a mathematically true statement. If your dosage calculation leads back to the correct total volume, you can proceed with confidence.

In healthcare, strict equality is rare. We care about vital signs staying within normal limits, blood sugar being below a threshold, or urine output being above a minimum volume.

An inequality is a mathematical statement comparing two values to show if one is less than, greater than, or not equal to another value.

• The symbol $<$ represents "less than".
• The symbol $>$ represents "greater than".
• The symbol $\le$ represents "less than or equal to".
• The symbol $\ge$ represents "greater than or equal to".

Solving an inequality utilizes the exact same properties of equality and inverse operations we just discussed, with one critical, vital exception.

Adding or subtracting the same value to both sides of an inequality does not change the direction of the inequality symbol. However, the logic of the number line flips when we introduce negative scaling. Multiplying or dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign to maintain a true statement.

Consider this carefully:

Multiplying both sides of the inequality $-2x < 4$ by $-0.5$ changes the inequality symbol to $>$, resulting in $x > -2$.

Source: Inversion of less-than-relation by multiplication with negative number by Stephan Kulla ( User:Stephan Kulla ), CC BY 3.0.

If you fail to flip the sign when multiplying or dividing by a negative, your entire clinical parameter reverses. You would interpret a minimum required dose as a maximum toxic dose!

Visualizing the Threshold on a Number Line

We graph inequalities on a horizontal number line to instantly see the entire set of possible solutions.

• An arrow pointing to the right on a horizontal number line represents values greater than a specific boundary number.
• An arrow pointing to the left on a horizontal number line represents values less than a specific boundary number.

But what about the boundary number itself?

• An open circle on a number line graph indicates that the boundary value is not included in the solution set of an inequality. Therefore, an open circle on a number line graph is used to represent "less than" strict inequalities ($<$) and "greater than" strict inequalities ($>$).
• A solid circle on a number line graph indicates that the boundary value is included in the solution set of an inequality. Consequently, a solid circle on a number line graph is used to represent "less than or equal to" ($\le$) and "greater than or equal to" ($\ge$) inequalities.

Math is simply a specialized language used to describe reality. Translating a word problem into an equation requires identifying the unknown quantity and assigning a letter variable to represent that quantity. Once you establish your $x$, you must decode the English phrasing into mathematical operators.

Here is your translation guide for the TEAS 7:

By breaking down the complex paragraphs of a clinical scenario or exam question into these structural elements, you transform a confusing block of text into a simple, solvable equation or inequality. You isolate the unknown, respect the balance of the operations, and ultimately arrive at truth—the precise calculation that guides your next action.