Finding the Truth in Mathematics
The Foundation: Understanding Equations
An equation is like a balanced scale. On the left side, you have an expression, and on the right side, you have another expression, and the equal sign $=$ signifies that both sides have the same value. The variable, often represented by a letter like $x$, is the unknown weight we need to find to keep the scale balanced.
For example, in the equation $x + 5 = 12$, we are looking for a number that, when added to 5, gives us 12. The solution is $x = 7$, because $7 + 5 = 12$ is a true statement.
Step-by-Step: Solving Linear Equations
A linear equation is an equation where the variable is raised only to the power of 1. The goal is to isolate the variable on one side of the equation.
Example 1: One-Step Equation
Solve $3x = 18$.
Since $x$ is multiplied by 3, we do the inverse operation to both sides: division.
$\frac{3x}{3} = \frac{18}{3}$
$x = 6$
Check: $3(6) = 18$. True!
Example 2: Two-Step Equation
Solve $2y - 7 = 15$.
Step 1: Undo the subtraction. Add 7 to both sides.
$2y - 7 + 7 = 15 + 7$ → $2y = 22$
Step 2: Undo the multiplication. Divide both sides by 2.
$\frac{2y}{2} = \frac{22}{2}$ → $y = 11$
Check: $2(11) - 7 = 22 - 7 = 15$. True!
Example 3: Equations with Variables on Both Sides
Solve $4x + 3 = 2x + 11$.
Step 1: Get all the variable terms on one side. Subtract $2x$ from both sides.
$4x - 2x + 3 = 2x - 2x + 11$ → $2x + 3 = 11$
Step 2: Isolate the variable term. Subtract 3 from both sides.
$2x + 3 - 3 = 11 - 3$ → $2x = 8$
Step 3: Solve for $x$. Divide both sides by 2.
$x = 4$
Check: $4(4)+3 = 16+3=19$ and $2(4)+11=8+11=19$. True!
Beyond the Line: Solving Quadratic Equations
A quadratic equation is an equation where the highest power of the variable is 2, generally in the form $ax^2 + bx + c = 0$. These equations can have zero, one, or two real solutions. The main methods for solving them are factoring, using the quadratic formula, and completing the square.
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Example 4: Solving by Factoring
Solve $x^2 - 5x + 6 = 0$.
Step 1: Factor the quadratic expression. We look for two numbers that multiply to 6 and add to -5. These are -2 and -3.
$(x - 2)(x - 3) = 0$
Step 2: Use the Zero Product Property, which states if a product is zero, then at least one of the factors must be zero.
$x - 2 = 0$ or $x - 3 = 0$
Step 3: Solve each simple equation.
$x = 2$ or $x = 3$
Check: For $x=2$: $(2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0$. True!
For $x=3$: $(3)^2 - 5(3) + 6 = 9 - 15 + 6 = 0$. True!
Example 5: Solving with the Quadratic Formula
Solve $2x^2 - 4x - 6 = 0$.
Here, $a = 2$, $b = -4$, $c = -6$.
Step 1: Calculate the discriminant[1]: $b^2 - 4ac = (-4)^2 - 4(2)(-6) = 16 + 48 = 64$.
Step 2: Plug into the formula.
$x = \frac{-(-4) \pm \sqrt{64}}{2(2)} = \frac{4 \pm 8}{4}$
Step 3: Solve for both possibilities.
$x = \frac{4 + 8}{4} = \frac{12}{4} = 3$ and $x = \frac{4 - 8}{4} = \frac{-4}{4} = -1$
The solutions are $x = 3$ and $x = -1$.
Not Just Equal: Solving Inequalities
Inequalities show a relationship where one expression is not necessarily equal to another, but rather greater than, less than, or equal to. The symbols used are $>$ (greater than), $<$ (less than), $\ge$ (greater than or equal to), and $\le$ (less than or equal to).
You solve inequalities much like you solve equations, with one critical rule:
Example 6: Simple Linear Inequality
Solve $3x + 7 \le 1$.
Step 1: Subtract 7 from both sides.
$3x \le -6$
Step 2: Divide both sides by 3 (a positive number, so the sign stays the same).
$x \le -2$
This means the solution is all real numbers less than or equal to -2.
Example 7: Inequality with Negative Division
Solve $-2x > 8$.
Step 1: Divide both sides by -2.
Step 2: Reverse the inequality sign because we are dividing by a negative number.
$x < -4$
This means the solution is all real numbers less than -4.
Putting Math to Work: Real-World Applications
Finding the value of a variable is not just an abstract exercise; it is used constantly in daily life and various professions.
Scenario 1: Budgeting and Shopping
Imagine you have $\$50$ and you want to buy as many $\$8$ books as possible, with at least $\$10$ left for bus fare. This translates to the inequality: $50 - 8b \ge 10$, where $b$ is the number of books.
Solving: $-8b \ge -40$ → $b \le 5$ (sign reversed!).
You can buy a maximum of 5 books.
Scenario 2: Geometry and Design
You need to build a rectangular garden with a length that is 5 meters more than its width. The area must be at least 84 square meters. If the width is $w$ meters, the length is $w+5$.
The area condition is: $w(w + 5) \ge 84$.
This becomes the quadratic inequality: $w^2 + 5w - 84 \ge 0$. Solving this (by first finding the roots of the corresponding equation) tells you the range of possible widths for your garden.
| Type | General Form | Key Method(s) | Solution Output |
|---|---|---|---|
| Linear Equation | $ax + b = c$ | Inverse Operations | A single number. |
| Quadratic Equation | $ax^2 + bx + c = 0$ | Factoring, Quadratic Formula | Zero, one, or two numbers. |
| Linear Inequality | $ax + b > c$ | Inverse Operations (Flip sign if ×/÷ by negative) | A range of numbers, often graphed on a number line. |
Common Mistakes and Important Questions
Q: Why do you flip the inequality sign when multiplying or dividing by a negative number?
Q: What does it mean if I get a statement that is never true, like $5 = 9$?
Q: What if I get a statement that is always true, like $0 = 0$ or $x = x$?
Footnote
[1] Discriminant: In the quadratic formula, the expression $b^2 - 4ac$. It determines the nature of the roots of a quadratic equation. If it is positive, there are two real solutions; if zero, one real solution; if negative, no real solutions.