Intro to Algebra: Expressions, Equations & Inequalities
Your first step into the exciting world of variables, symbols, and solving puzzles!
Welcome to Algebra! 🚀
Algebra is a powerful tool that uses letters (like x or y) to represent unknown numbers. It helps us solve puzzles and describe relationships. In this lesson, we’ll learn about the three basic building blocks of algebra: expressions, equations, and inequalities.
🧩 What is an Algebraic Expression?
An algebraic expression is a mathematical phrase that combines numbers, variables, and operators (like +, −, ×, ÷). Think of it as a recipe for a calculation. Crucially, an expression does not have an equals sign.
What’s a variable? A variable is a letter that stands in for a number we don’t know yet. The most common variables are x and y, but you can use any letter!
Examples of expressions:
- $x + 5$ (5 more than an unknown number)
- $3y – 2$ (3 times a number, then subtract 2)
- $[[frac{n}{4}]]$ (an unknown number divided by 4)
Structure of an Expression
An expression is made of terms. For example, in the expression $2x + 7$:
- $2x$ is a variable term.
- $7$ is a constant term.
- $+$ is the operator.
$$[[text{Variable Term}]] + [[text{Constant Term}]]$$
Translating Words into Expressions
Algebra helps us turn word problems into math. Let’s see how.
- “A number increased by 8” becomes $n + 8$
- “The product of 7 and a number” becomes $7y$
- “10 less than a number” becomes $x – 10$
🎯 What is an Equation?
An equation is a statement that two expressions are equal. It’s like a balanced scale. You’ll always see an equals sign (=) in an equation. The goal is usually to “solve” the equation, which means finding the value of the variable that makes the statement true.
Expression vs. Equation
- $x + 3$ is an expression (a phrase).
- $x + 3 = 10$ is an equation (a complete sentence stating two things are equal).
In the equation $x + 3 = 10$, we can figure out that $x$ must be 7 to make the scale balance.
Solving a Simple Equation
To solve an equation, we need to get the variable by itself on one side of the equals sign. Whatever you do to one side, you must do to the other to keep it balanced.
Let’s solve for $x$ in the equation $x + 4 = 9$.
Our goal is to get $x$ alone. To undo the “+ 4”, we subtract 4.
$$x + 4 – 4 = 9 – 4$$
Now, we simplify both sides.
$$x = 5$$
We found the solution! If we plug 5 back into the original equation, we get $5 + 4 = 9$, which is true.
The Golden Rule of Equations
To keep an equation balanced, whatever operation you perform on one side of the equals sign, you must also perform on the other side.
$$[[text{If }]] A = B, [[text{ then }]] A+c = B+c$$
⚖️ Understanding Inequalities
An inequality is a statement that compares two expressions that are not necessarily equal. It tells us if one value is greater than, less than, or not equal to another.
Inequality Symbols
- $ > $ : greater than
- $ < $ : less than
- $ [[geq]] $ : greater than or equal to
- $ [[leq]] $ : less than or equal to
- $ [[neq]] $ : not equal to
Memory Tip: The alligator’s mouth always wants to eat the bigger number! $5 > 2$.
Real-World Inequalities
- To ride a rollercoaster, your height (h) must be at least 1.4 metres: $h [[geq]] 1.4$
- The maximum capacity (c) of an elevator is 10 people: $c [[leq]] 10$
- To pass the test, your score (s) must be greater than 80%: $s > 80$
Structure of an Inequality
An inequality compares two expressions using one of the inequality symbols.
$$x + 3 [[leq]] 10$$
This reads as “x plus 3 is less than or equal to 10.” This means $x$ could be 7, 6, 5, or any number smaller than 7.
Practice Problems
Test your knowledge with these practice problems. Check your answers to see how you did!
1. Translate the following phrase into an algebraic expression: "A number n decreased by 7."
Show Answer
The phrase "decreased by 7" means we need to subtract 7.
The expression is: $n - 7$
2. Solve the equation: $y - 5 = 11$.
Show Answer
To solve for y, we need to get it by itself. We can undo the "- 5" by adding 5 to both sides of the equation.
$$y - 5 + 5 = 11 + 5$$
Simplifying both sides gives us the answer.
$$y = 16$$
3. Which symbol represents "at least" or "greater than or equal to"?
Show Answer
The symbol for "greater than or equal to" is $ [[geq]] $.
4. You need at least 50 points (p) to win the game. Write this as an inequality.
Show Answer
"At least 50" means you can have 50 points or more. This means the number of points (p) must be greater than or equal to 50.
The inequality is: $p [[geq]] 50$
Interactive Quiz
Ready to check your understanding? Take this short quiz!