Numbers & Operations

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Recognize and classify key number types (integer, rational, irrational, real).
  • Apply order of operations accurately.
  • Use commutative, associative, and distributive properties to simplify expressions.
  • Work with absolute value using distance-based reasoning.
  • Solve basic inequalities, including sign changes when negatives are involved.

Key Ideas

Number Types

Let’s organize the main categories of numbers:

  • Whole numbers: \(0, 1, 2, 3, \ldots\)
  • Integers: include whole numbers and negatives
  • Rational numbers: can be written as \(\frac{p}{q}\) for integers \(p, q\neq 0\)
  • Irrational numbers: cannot be expressed as a simple fraction (e.g., \(\sqrt{2}\), \(\pi\))
  • Real numbers: all rational and irrational numbers combined

Quick check to think about:
Is \(\sqrt{50}\) a rational number? Keep that in mind as you go.

Order of Operations

Remember:
Parentheses → Exponents → Multiplication/Division → Addition/Subtraction
…but multiplication and division happen left-to-right, as do addition and subtraction.

Example:
\[ 3 + 2 \cdot 5^2 = 3 + 2 \cdot 25 = 3 + 50 = 53 \]

Properties of Operations

These help you reorganize expressions safely:

  • Commutative: \(a+b=b+a\) and \(ab=ba\)
  • Associative: \((a+b)+c = a+(b+c)\)
  • Distributive: \(a(b+c)=ab+ac\)

Absolute Value

\(|x|\) is the distance from 0 on a number line, so it is always non-negative.

Examples:
\(|5| = 5\)
\(|-7| = 7\)

Inequalities and Sign Flips

When multiplying or dividing an inequality by a negative, flip the inequality sign.

Example:
\(-3x > 12\) → divide by \(-3\) → \(x < -4\)

Important

If a negative number multiplies or divides the variable, pause and check whether the inequality should flip.

Common Problem Types

Evaluating Expressions

Use parentheses and PEMDAS carefully.

Example:
Evaluate \(4 - 3^2 + 2\).
\(4 - 9 + 2 = -3\).

Classifying Numbers

Ask yourself: Can it be written as a fraction? Can it be simplified?

Example:
\(\sqrt{50} = 5\sqrt{2}\) → irrational.

Simplifying Expressions

Distribute first, then combine like terms.

Working with Absolute Value

Example:
\(|3 - 8| = |-5| = 5\).

Solving Basic Inequalities

Example:
\(5 - 2x \le 11\)
\(-2x \le 6\)
\(x \ge -3\)

Strategies

  • If uncertain whether a number is rational, try writing it as a fraction.
  • Always simplify expressions under radicals.
  • Sketch quick number lines for absolute value problems.
  • Rewrite subtraction as \(+(-)\) to reduce sign mistakes.
  • Track negatives carefully when working with inequalities.

Worked Examples

Example 1

Simplify:
\(4(3 - x) + 2x\)

Step 1: Distribute the 4
\(4 \cdot 3 = 12\), \(4 \cdot (-x) = -4x\)
So the expression becomes:
\(12 - 4x + 2x\)

Step 2: Combine like terms
\(-4x + 2x = -2x\)

Answer:
\(12 - 2x\)

Example 2

Classify: Is \(\frac{2}{7}\) rational, irrational, or an integer?

Step 1: A rational number can be written as a ratio of integers.
\(\frac{2}{7}\) is already a ratio of integers.

Step 2: It is not a whole number or integer.

Answer:
Rational.

Example 3

Solve: \(|x - 3| = 5\)

Step 1: Set up the two cases:
1) \(x - 3 = 5\)
2) \(x - 3 = -5\)

Step 2: Solve each:
Case 1: \(x = 8\)
Case 2: \(x = -2\)

Answer:
\(x = 8\) or \(x = -2\)

Example 4

Solve: \(-4x > 20\)

Step 1: Divide both sides by \(-4\)
Remember: dividing by a negative flips the inequality.

Step 2:
\(x < -5\)

Answer:
\(x < -5\)

Practice Problems

  1. Simplify: \(7 - 2(3 - x)\)
  2. Classify: \(\sqrt{16}\)
  3. Evaluate: \(| -12 + 5 |\)
  4. Solve: \(3x - 4 < 11\)
  5. Is \(\frac{4}{9}\) rational?

1. \(7 - 2(3 - x)\)
Distribute: \(7 - 6 + 2x\)
Combine: \(1 + 2x\)

2. \(\sqrt{16} = 4\) → rational

3. \(|-12 + 5| = |-7| = 7\)

4. \(3x - 4 < 11\)
Add 4: \(3x < 15\)
Divide: \(x < 5\)

5. \(\frac{4}{9}\) is rational (a ratio of integers)

WarningCommon Mistakes to Avoid
  • Forgetting to flip the inequality sign when dividing by a negative
  • Dropping negative signs while simplifying
  • Treating MD or AS as separate steps instead of left-to-right
  • Not simplifying radicals before classifying rational/irrational numbers

Summary

  • Numbers fall into main categories: whole, integer, rational, irrational, real.
  • Apply order of operations strictly with left-to-right rules for MD and AS.
  • Use algebraic properties to simplify expressions.
  • Absolute value represents distance and is always non-negative.
  • Inequalities may require a sign flip when multiplying or dividing by negatives.
  • Multiplication/division and addition/subtraction both follow left-to-right rules.
  • Square roots of perfect squares are rational.
  • Watch for hidden negatives when solving inequalities.
  • Absolute value equations typically produce two solutions.