3.2 Add Integers

Integer addition is a fundamental concept that builds on what you already know about whole numbers. By introducing negative numbers, you can now represent real-world situations like temperature drops, debt, and elevation changes with math. This section covers how to add integers using number lines and rules, how to evaluate expressions, and how to translate word problems into integer addition.

Understanding Integer Addition

Number lines for integer addition

A number line is the best way to see what's happening when you add integers. Zero sits in the middle, positive integers stretch to the right, and negative integers stretch to the left.

Here's how to use it:

1. Start at the first number on the number line.
2. If you're adding a positive integer, move to the right by that many spaces.
3. If you're adding a negative integer, move to the left by that many spaces.
4. Where you land is your answer.

For example, to solve $-3 + 5$: start at $-3$, move 5 spaces to the right, and you land on $2$. To solve $4 + (-6)$: start at $4$, move 6 spaces to the left, and you land on $-2$.

Simplification of integer expressions

Once you're comfortable with the number line, you can use two rules to add integers without drawing anything:

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

• $5 + 3 = 8$ (both positive, so the answer is positive)
• $-5 + (-3) = -8$ (both negative, so the answer is negative)

Different signs: Subtract the smaller absolute value from the larger one, then use the sign of the number with the larger absolute value.

• $-7 + 4 = -3$ (7 is larger than 4, and 7 is negative, so the answer is negative)
• $9 + (-6) = 3$ (9 is larger than 6, and 9 is positive, so the answer is positive)

You can also apply these rules when simplifying longer expressions. For instance, $2x - 3 + 4x + 5$ simplifies to $6x + 2$ by combining like terms: $2x + 4x = 6x$ and $-3 + 5 = 2$.

Evaluation of variable expressions

To evaluate an expression, you replace the variable with a given number and then simplify using the integer addition rules.

For example, if $x = -2$, evaluate $3x + 4$:

1. Substitute: $3(-2) + 4$
2. Multiply: $-6 + 4$
3. Apply the different-signs rule: $|-6| = 6$ and $|4| = 4$, so subtract to get $2$, and use the negative sign since 6 is larger. The result is $-2$.

Word problems to algebraic expressions

Turning a word problem into an expression takes practice. Here's a reliable approach:

1. Identify the unknown quantity and assign it a variable (e.g., let $x$ represent the number).
2. Look for keywords: "more than," "gained," or "increased" signal positive integers; "less than," "lost," or "decreased" signal negative integers.
3. Write the expression. For example, "5 more than twice a number" becomes $2x + 5$.
4. Simplify by combining like terms if possible.

Real-world applications of integers

• Temperature changes: A temperature increase of 10°C is $+10$, and a decrease of 5°C is $-5$. Adding them gives the net change: $10 + (-5) = 5$°C.
• Financial transactions: Income of $500 is $+500$, and an expense of $250 is $-250$. The net result: $500 + (-250) = 250$ dollars.
• Elevation changes: Climbing 200 ft is $+200$, and descending 150 ft is $-150$. Net change: $200 + (-150) = 50$ ft.

Applying Integer Addition Concepts

Solve real-world problems using integer addition

1. Read the problem carefully and identify the given information (e.g., the temperature rose 7°C then dropped 3°C).

2. Determine what you need to find (the net temperature change).

3. Translate into an integer expression: $+7 + (-3)$.

4. Simplify using the rules: $7 - 3 = 4$.

5. Interpret the result: the temperature is 4°C higher than where it started.

Key Concepts in Integer Addition

• Integers: The set of whole numbers and their negatives, including zero: $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$
• Absolute value: The distance a number is from zero on the number line, always positive (e.g., $|-7| = 7$)
• Addends: The numbers being added together
• Sum: The result of an addition operation
• Identity property of zero: Adding 0 to any integer leaves it unchanged ($a + 0 = a$)