ACT Math: Pre-Algebra and Elementary Algebra — Numbers, Equations, Inequalities

TL;DR

This section covers foundational math skills like understanding number properties, solving basic equations, and working with inequalities, which are essential for most ACT Math problems. You'll master operations with various number types, isolate variables in equations, and determine valid ranges for solutions using inequalities. Focus on simplifying expressions and applying inverse operations to efficiently solve these core problem types.

1. The Mental Model

Think of numbers, equations, and inequalities as the building blocks of math. Numbers are your quantities, equations are balanced scales showing equality, and inequalities are unbalanced scales showing one side is bigger or smaller.

2. The Core Material

This topic focuses on the fundamental concepts that form the basis of most ACT Math problems. You'll need to be quick and accurate with these.

Numbers: Properties and Operations

The ACT tests your understanding of different number types and how they behave.

• Integers: Whole numbers, positive, negative, or zero (..., -2, -1, 0, 1, 2, ...).
• Rational Numbers: Numbers that can be expressed as a fraction a/b, where a and b are integers and b ≠ 0. This includes integers, fractions, and terminating or repeating decimals.
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Their decimal representation is non-terminating and non-repeating (e.g., π, √2).
• Real Numbers: All rational and irrational numbers. This is what you usually deal with.

You need to know basic operations (addition, subtraction, multiplication, division) and how they apply to positive and negative numbers. Remember the order of operations (PEMDAS/BEDMAS).

Properties you should know:
* Commutative Property: Order doesn't matter for addition (a + b = b + a) or multiplication (a × b = b × a).
* Associative Property: Grouping doesn't matter for addition (a + (b + c) = (a + b) + c) or multiplication (a × (b × c) = (a × b) × c).
* Distributive Property: a × (b + c) = (a × b) + (a × c).

Equations: Solving for Variables

An equation states that two expressions are equal. Your goal is usually to find the value(s) of the variable(s) that make the equation true.

The key principle: Whatever you do to one side of an equation, you must do to the other side to maintain balance.

Steps to solve basic linear equations:
1. Combine like terms on each side of the equation.
2. Isolate the variable term by adding or subtracting constants from both sides.
3. Isolate the variable by multiplying or dividing both sides by the coefficient of the variable.

Example: 3x - 5 = 10
* Add 5 to both sides: 3x = 15
* Divide by 3 on both sides: x = 5

For equations with variables on both sides, first get all variable terms to one side and all constant terms to the other.

Inequalities: Understanding Ranges

Inequalities compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).

Solving inequalities is very similar to solving equations, with one crucial difference:
When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.

Example: -2x < 6
* Divide by -2 (and reverse the sign): x > -3

This means any number greater than -3 will satisfy the original inequality.

Here's a quick look at the process for solving inequalities:

graph TD
    A["Start with Inequality"] --> B{"Isolate Variable Term?"}
    B -- Yes --> C{"Isolate Variable?"}
    B -- No --> D["Combine Like Terms"] --> B
    C -- Divide/Multiply by Negative --> E["Reverse Inequality Sign"] --> F["Solution (e.g., x > 5)"]
    C -- Divide/Multiply by Positive --> F
    C -- Add/Subtract --> F

3. Worked Example

Let's solve the inequality 5 - 2(x + 3) ≥ 11.

1. Distribute: Apply the -2 to the terms inside the parentheses.
   5 - 2x - 6 ≥ 11

2. Combine like terms on the left side (5 and -6).
   -1 - 2x ≥ 11

3. Isolate the variable term: Add 1 to both sides.
   -2x ≥ 12

4. Isolate the variable: Divide both sides by -2. Remember to flip the inequality sign!
   x ≤ -6

So, the solution is x ≤ -6.

4. Key Takeaways

• Master the order of operations (PEMDAS/BEDMAS) for all calculations.
• Understand the properties of numbers (commutative, associative, distributive) to simplify expressions efficiently.
• To solve equations, always perform the same operation on both sides to maintain equality.
• When solving inequalities, remember to reverse the inequality sign if you multiply or divide by a negative number.
• Practice translating word problems into equations or inequalities.
• Double-check your arithmetic, especially with positive and negative numbers.
• Know that integers are whole numbers, rational numbers can be fractions, and irrational numbers cannot.

5. Now Try It

Spend 15 minutes solving the following problems without a calculator, focusing on accuracy and showing your steps:

1. Solve for x: 4(x - 2) + 7 = 3x - 1
2. Solve for y: -3y + 10 < 2y - 5
3. Simplify the expression: (8 - 5 × 2) ÷ (-2)
4. If a = -3 and b = 4, evaluate 2a² - 3b.

What success looks like: You should be able to arrive at the correct solution for each problem, clearly showing the steps you took. For the inequality, make sure you applied the sign-flipping rule correctly if needed. For simplification, get the correct single numeric answer.