1.3 Add and Subtract Integers

Understanding Integers and Their Operations

Integers include all positive whole numbers, negative whole numbers, and zero. They show up constantly in real life (temperatures, bank accounts, elevations) and form the backbone of everything you'll do in algebra. This section covers what integers are, how absolute value works, and how to add and subtract them confidently.

Negative Numbers in Context

Negative numbers represent quantities below zero or in an opposite direction from positive values.

• Temperature: $-5°C$ means 5 degrees below the freezing point of water
• Elevation: $-50$ m means 50 meters below sea level (like Death Valley)
• Finance: $-100$ dollars means a debt or withdrawal of $100$

Every number has an opposite, which sits the same distance from zero but on the other side of the number line. The opposite of $7$ is $-7$, and the opposite of $-3$ is $3$. Zero is its own opposite.

Absolute Value

Absolute value measures how far a number sits from zero on the number line, regardless of direction. It's written with vertical bars: $|x|$.

• $|8| = 8$ because 8 is 8 units from zero
• $|-4| = 4$ because $-4$ is also 4 units from zero
• $|0| = 0$

Absolute value always gives a result that is zero or positive. You'll use it frequently when deciding the sign of an answer in addition and subtraction problems.

Performing Operations with Integers

Adding Integers

There are two cases to remember, depending on whether the signs match.

Case 1: Same signs. Add the absolute values and keep the shared sign.

• $(+9) + (+5) = +14$
• $(-6) + (-3) = -9$

Case 2: Different signs. Subtract the smaller absolute value from the larger one. The answer takes the sign of whichever number had the larger absolute value.

• $(+7) + (-4) = +3$ (because $7 > 4$, and $7$ is positive)
• $(-8) + (+2) = -6$ (because $8 > 2$, and $8$ is negative)

One useful fact: any number plus its opposite equals zero. This is the additive inverse property: $a + (-a) = 0$.

Subtracting Integers

The core idea is that subtraction is the same as adding the opposite. For any integers $a$ and $b$:

$a - b = a + (-b)$

Here's how to apply it step by step:

1. Rewrite the subtraction as addition of the opposite.
2. Follow the addition rules from above (same signs or different signs).
3. Simplify.

Subtracting a positive is the same as adding a negative:

• $10 - 7 = 10 + (-7) = 3$

Subtracting a negative is the same as adding a positive (the two negatives "cancel"):

• $-5 - (-8) = -5 + 8 = 3$

More examples:

• $(+12) - (+9) = 12 + (-9) = +3$
• $(-4) - (+6) = -4 + (-6) = -10$
• $(+3) - (-11) = 3 + 11 = +14$

A common mistake is forgetting to flip the sign of the second number when you rewrite subtraction as addition. Always change both the operation and the sign of the number that follows it.

Integer Arithmetic and Algebraic Expressions

Order of Operations

When an expression has more than one operation, you need to evaluate it in the correct order. The standard rule is PEMDAS:

1. Parentheses (work inside them first)
2. Exponents
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

For example, in $3 + 2 \times (-4)$, you multiply first: $2 \times (-4) = -8$, then add: $3 + (-8) = -5$.

Signed Numbers

Integers are just one type of signed number. The same sign rules you learned here for addition and subtraction also apply to fractions and decimals. Once you're comfortable with integers, extending to $-2.5 + 1.3$ or $-\frac{1}{2} - \frac{3}{4}$ uses the exact same logic.

Algebraic Expressions

An algebraic expression combines variables, constants, and operations. For example, $2x - 3y + 5$ contains three terms: $2x$, $-3y$, and $5$. The integer arithmetic you've practiced here is exactly what you'll use to simplify and evaluate expressions like these once you plug in values for the variables.