Linear Equations & Inequalities | Math Learning Guide

📋 Prerequisites

Variables & Simple Equations

Linear equations are the workhorse of algebra. Every competition has problems that boil down to setting up and solving a linear equation — or inequality. Master this and you've unlocked a huge chunk of algebra, geometry, and even number theory problems.

📚 Key Concepts

Standard Five-Step Method

For any linear equation $ax + b = cx + d$:

Distribute if there are parentheses
Combine like terms on each side
Collect all $x$ terms on one side (move the smaller coefficient to avoid negatives)
Collect all constants on the other side
Divide by the coefficient of $x$

Always check your answer by plugging back into the original equation!

Equations with Variables on Both Sides

The key rule: move terms by doing the opposite operation to both sides.

Example: $ 3x + 5 = x + 11 $

Subtract $x$ from both sides: $ 2x + 5 = 11 $
Subtract 5 from both sides: $ 2x = 6 $
Divide by 2: $ x = 3 $

Linear Inequalities

Solve just like equations, with one critical exception:

⚠️ The One Rule: When you multiply or divide both sides by a negative number, you must flip the inequality sign.

Example: $ -2x < 6 $ \\to divide by -2, flip sign \\to $ x > -3 $

Notation:

$ < $ and $ > $ — open circle on number line, parentheses in interval notation
$ \le $ and $ \ge $ — closed (filled) circle, brackets in interval notation

Word Problem Types

Three classic types show up constantly:

Distance / Rate: $ \text{distance} = \text{rate} \times \text{time} $ ($d = rt$)
Work / Rate: Combined rate = sum of individual rates
Mixture / Concentration: $Amount of pure stuff = concentration \\times total volume$

For all word problems: define your variable clearly, write what you know, translate to an equation, solve, and check that your answer makes sense in context.

Special Cases

No solution: $ 2x + 3 = 2x + 5 $ $\to$ $ 3 = 5 $ (impossible). Parallel lines.
Infinite solutions: $ 2(x+1) = 2x + 2 $ $\to$ $ 2x+2 = 2x+2 $ (always true). Same line.
Identity: An equation true for all values of $x$

💡 Key Insight: When solving word problems, always ask "what am I being asked for?" — that's your variable. Then ask "what relationships do I know?" — those are your equations. If you can write down what you know in plain English first, translating to math is much easier.

✏️ Example Problems

📝 Example 1 (Variables on both sides)

Solve for $x$: $ 4x - 7 = 2x + 15 $

Step 1: Get $x$ terms on the left, constants on the right.

Subtract $2x$ from both sides:

\[ 2x - 7 = 15 \]

Step 2: Add 7 to both sides.

\[ 2x = 22 \]

Step 3: Divide by 2.

\[ x = 11 \]

Check: Left: $ 4(11) - 7 = 44 - 7 = 37 $. Right: $ 2(11) + 15 = 22 + 15 = 37 $. ✓

Answer: 11

📝 Example 2 (Inequality with negative coefficient)

Solve: $ -3x + 4 \ge 13 $

Step 1: Subtract 4 from both sides.

\[ -3x \ge 9 \]

Step 2: Divide both sides by -3. Remember to flip the inequality sign!

\[ x \le -3 \]

Check: Try x = -3: $ -3(-3) + 4 = 9 + 4 = 13 \ge 13 $. ✓

$Try x = -4 (which is \\leq -3):$ $ -3(-4) + 4 = 12 + 4 = 16 \ge 13 $. ✓

$Try x = 0 (which is NOT \\leq -3):$ $ 0 + 4 = 4 \ge 13 $? No. ✓

Answer: $ x \le -3 $ (boundary at -3)

📝 Example 3 (Distance word problem)

Two cars leave the same town at the same time, traveling in opposite directions. One goes 50 mph, the other 60 mph. After how many hours are they 330 miles apart?

Step 1: Let $t$ = number of hours traveled.

Step 2: Use $ d = rt $ for each car.

Car 1 distance: $ 50t $ miles
Car 2 distance: $ 60t $ miles

Step 3: They're going opposite directions, so total distance apart = sum of both distances.

\[ 50t + 60t = 330 \]

Step 4: Solve.

\[ 110t = 330 \\ t = 3 \]

Check: In 3 hours, Car 1 goes 150 mi, Car 2 goes 180 mi. Total apart: 150 + 180 = 330 mi. ✓

Answer: 3 hours