Linear Equations & Inequalities

From calculating how much money you need to save each month to figuring out the break-even point for a small business -- linear equations and inequalities are tools you will use throughout your life. Let us sharpen these skills before Grade 11.

Solving Linear Equations

A linear equation has the variable raised to the power of 1 (like 2x + 3 = 7, not x^2).

The Golden Rule: Whatever you do to one side of the equation, you must do to the other side.

Worked Example: Solve 3x - 7 = 2x + 5

Step 1: Move variable terms to one side: 3x - 2x = 5 + 7

Step 2: Simplify: x = 12

Worked Example with fractions: Solve (x + 2)/3 = (x - 1)/2

Step 1: Cross-multiply: 2(x + 2) = 3(x - 1)

Step 2: Expand: 2x + 4 = 3x - 3

Step 3: Solve: 4 + 3 = 3x - 2x, so x = 7

Linear Inequalities

Inequalities use symbols like < (less than), > (greater than), and their "or equal to" versions.

Critical Rule: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

Worked Example: Solve 2x - 3 > 7

Step 1: Add 3 to both sides: 2x > 10

Step 2: Divide by 2: x > 5

Worked Example (with sign flip): Solve -3x + 6 ≤ 15

Step 1: Subtract 6: -3x ≤ 9

Step 2: Divide by -3 (flip the sign!): x ≥ -3

Graphing on the Number Line

• For x > 5: draw an open circle at 5 and shade to the right
• For x ≥ -3: draw a filled (closed) circle at -3 and shade to the right
• For x < 2: draw an open circle at 2 and shade to the left

Open circle = value NOT included. Filled circle = value IS included.

Nepal Connection: If a student needs at least 40 marks out of 100 to pass the SEE exam, this can be written as the inequality m ≥ 40. Inequalities help us express real-world conditions mathematically.

Key Takeaways

• Solve equations by isolating the variable using inverse operations
• With inequalities, flip the sign when multiplying or dividing by a negative
• Open circles on number lines mean "not included"; filled circles mean "included"
• Always check your answer by substituting back into the original equation