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

TL;DR

You'll tackle questions involving basic math operations, understanding number properties, and solving for unknowns in equations and inequalities. Focus on simplifying expressions, isolating variables, and knowing when to flip inequality signs. Practice is key to quickly identifying the best approach for each problem type.

1. The Mental Model

Think of these questions as puzzles where you need to find a missing piece using fundamental math rules. It's about breaking down problems into smaller steps, much like following a recipe, to get to the correct answer. Your goal is to apply the right operation at the right time.

2. The Core Material

This section covers the foundational math skills you'll need for many ACT problems. It's all about numbers, how they relate, and how you manipulate them.

a. Number Properties and Operations

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You need to be solid on integers (whole numbers, positive and negative, including zero), rational numbers (numbers that can be written as a fraction, like 0.5 or -3), and irrational numbers (numbers that can't be written as a fraction, like $\sqrt{2}$ or $\pi$). Remember the order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)).

Example: $5 + 3 \times (8 - 2) \div 2 = 5 + 3 \times 6 \div 2 = 5 + 18 \div 2 = 5 + 9 = 14$

b. Solving Equations

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Your main goal when solving an equation is to isolate the variable (the unknown, usually 'x'). Whatever you do to one side of the equation, you must do to the other side to keep it balanced.

• Addition/Subtraction: To undo addition, subtract. To undo subtraction, add.
  Example: $x + 7 = 10 \Rightarrow x = 10 - 7 \Rightarrow x = 3$
• Multiplication/Division: To undo multiplication, divide. To undo division, multiply.
  Example: $3x = 15 \Rightarrow x = 15 / 3 \Rightarrow x = 5$
• Multi-step Equations: Combine like terms first, then work to isolate the variable using inverse operations.
  Example: $2x + 5 = 11 \Rightarrow 2x = 11 - 5 \Rightarrow 2x = 6 \Rightarrow x = 3$

c. Solving Inequalities

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Inequalities ($<, >, \le, \ge$) are very similar to equations. You solve them the same way, with one crucial difference:

• Flipping the Sign: If you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign.

Here's how you decide whether to flip the sign:

graph TD
    Start["Begin Solving Inequality"] --> A{Multiply or Divide by a Number?};
    A -- No --> B["Solve like an equation (no sign flip)"];
    A -- Yes --> C{Is the Number Negative?};
    C -- No --> B;
    C -- Yes --> D["Multiply/Divide AND FLIP Inequality Sign!"];
    B --> End["Solution Found"];
    D --> End;

Example 1: $x - 3 > 5 \Rightarrow x > 5 + 3 \Rightarrow x > 8$ (No sign flip)
Example 2: $-2x \ge 10 \Rightarrow x \le 10 / (-2) \Rightarrow x \le -5$ (Sign flipped because we divided by -2)

d. Word Problems

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Many ACT questions are word problems. The trick is to translate the words into mathematical expressions, equations, or inequalities. Look for keywords:
* "is," "was," "equals" $\rightarrow =$
* "sum," "more than," "increased by" $\rightarrow +$
* "difference," "less than," "decreased by" $\rightarrow -$
* "product," "times," "of" $\rightarrow \times$
* "quotient," "per," "divided by" $\rightarrow \div$

3. Worked Example

Let's solve a multi-step inequality word problem:

"Three less than twice a number is at most 15. What are the possible values for the number?"

1. Translate "a number": Let's call it $x$.
2. Translate "twice a number": This is $2x$.
3. Translate "Three less than twice a number": This means $2x - 3$. (Be careful not to write $3 - 2x$!).
4. Translate "is at most 15": This means it's less than or equal to 15, so $\le 15$.

Putting it all together, the inequality is: $2x - 3 \le 15$

Now, solve it:
* Add 3 to both sides: $2x - 3 + 3 \le 15 + 3$
$2x \le 18$
* Divide both sides by 2: $2x / 2 \le 18 / 2$
$x \le 9$

The possible values for the number are any number less than or equal to 9.

4. Key Takeaways

• Always follow the order of operations (PEMDAS) precisely to avoid calculation errors.
• When solving equations, whatever you do to one side, you must do to the other.
• For inequalities, remember to flip the sign ONLY when multiplying or dividing by a negative number.
• Practice translating word problems into mathematical expressions; this is a critical skill.
• Double-check your arithmetic, especially with negative numbers and fractions.

Common Mistakes to Avoid:
- Not flipping the inequality sign when necessary.
- Incorrectly applying the order of operations, especially with subtraction and division.
- Misinterpreting "less than" in word problems (e.g., $3 - x$ instead of $x - 3$).
- Distributing a negative sign incorrectly inside parentheses.

5. Now Try It

Spend 15 minutes working through a mix of 5-7 practice problems that involve:
1. Simplifying an expression with multiple operations (PEMDAS).
2. Solving a two-step linear equation.
3. Solving a linear inequality that requires flipping the sign.
4. Solving a linear inequality that doesn't require flipping the sign.
5. Translating and solving a simple word problem involving an equation or inequality.

Success looks like: You can consistently and accurately arrive at the correct solution for each type of problem, identifying when to flip an inequality sign without hesitation.