Introduction to Linear Equations and Inequalities

Welcome to one of the most important chapters in GCSE Mathematics! In this section, we are going to learn how to find "the unknown." Whether you are trying to figure out the cost of a single item in a multi-pack or calculating how much time you have left to finish a race, you are using the logic of linear equations.

An equation is like a balanced set of scales. If the scales are balanced, both sides must have the exact same value. Our goal is to move things around until we find out exactly what value our "unknown" (usually the letter \(x\)) represents.

An inequality is slightly different—it tells us that one side is larger or smaller than the other. Instead of one single answer, we often find a range of possible answers.


1. Solving Linear Equations

To solve a linear equation, we use inverse operations. Think of this as "undoing" the math to get the letter by itself.

The Golden Rule of Algebra

Whatever you do to one side of the equation, you MUST do to the other side. This keeps the "scales" balanced.

Quick Review of Inverse Operations:

  • The inverse of addition \(+\) is subtraction \(-\).
  • The inverse of subtraction \(-\) is addition \(+\).
  • The inverse of multiplication \(\times\) is division \(\div\).
  • The inverse of division \(\div\) is multiplication \(\times\).

Solving with the Unknown on One Side

Example: Solve \(4x - 7 = 13\)

  1. Step 1: Undo the "subtract 7" by adding 7 to both sides.
    \(4x = 20\)
  2. Step 2: Undo the "multiply by 4" by dividing both sides by 4.
    \(x = 5\)

Solving with the Unknown on Both Sides

If you see an \(x\) on both sides, don't panic! Your first goal is to move all the \(x\) terms to one side (usually the side that has more of them).

Example: Solve \(5x + 3 = 2x + 15\)

  1. Step 1: Subtract \(2x\) from both sides to get all \(x\)'s on the left.
    \(3x + 3 = 15\)
  2. Step 2: Subtract 3 from both sides.
    \(3x = 12\)
  3. Step 3: Divide by 3.
    \(x = 4\)

Common Mistake to Avoid: When you move a term, make sure you perform the opposite operation. If it is positive, subtract it; if it is negative, add it!


2. Dealing with Brackets and Fractions

Sometimes, equations look a bit messy because they have brackets or fractions. We need to "clean them up" first.

Equations with Brackets

If you see brackets, expand them first (multiply the term outside the bracket by everything inside).
Example: Solve \(3(x + 4) = 21\)
Expand: \(3x + 12 = 21\)
Subtract 12: \(3x = 9\)
Divide by 3: \(x = 3\)

Equations with Fractions

To get rid of a fraction, multiply the entire equation by the denominator (the bottom number).
Example: Solve \(\frac{x}{5} + 2 = 6\)
Subtract 2 first: \(\frac{x}{5} = 4\)
Multiply by 5: \(x = 20\)

Key Takeaway: Always aim to simplify the equation into the form \(ax = b\) or \(ax + b = c\) before doing the final division.


3. Translating Words into Equations

In the exam, you might be asked to "form an equation" from a story.
"I think of a number, multiply it by 3 and add 5. The answer is 26."

Let the number be \(x\).
Multiply by 3: \(3x\)
Add 5: \(3x + 5\)
The answer is 26: \(3x + 5 = 26\)
Now you can solve it! \(3x = 21 \implies x = 7\).


4. Linear Inequalities

Inequalities use different symbols instead of an equals sign:

  • \(<\) : Less than
  • \(>\) : Greater than
  • \(\le\) : Less than or equal to
  • \(\ge\) : Greater than or equal to

Solving Inequalities

You solve a linear inequality exactly like an equation. The only difference is that you keep the inequality symbol throughout.

Example: Solve \(2x - 3 < 7\)
Add 3: \(2x < 10\)
Divide by 2: \(x < 5\)

Representing Inequalities on a Number Line

Because there are many answers (any number less than 5), we draw them on a number line:

  • Use an Open Circle \(\circ\) for \(<\) or \(>\) (this means the number itself is not included).
  • Use a Closed/Solid Circle \(\bullet\) for \(\le\) or \(\ge\) (this means the number is included).
  • Draw an arrow in the direction of the possible values.

Did you know? If you multiply or divide an inequality by a negative number, you must flip the inequality sign! For example, \(-2x < 10\) becomes \(x > -5\).


5. Higher Tier Topics (H)

If you are taking the Foundation tier, you can skip this section!

Inequalities in Two Variables and Shaded Regions

You can represent inequalities on a graph. For example, \(y > 2x + 1\):

  1. Draw the line \(y = 2x + 1\).
  2. Use a dashed line for \(<\) or \(>\).
  3. Use a solid line for \(\le\) or \(\ge\).
  4. Shade the region that satisfies the inequality. For \(y > \dots\), shade above the line. For \(y < \dots\), shade below the line.

Set Notation

Higher tier students should recognise set notation for inequalities.
The expression \(\{x : x > 3\}\) simply means "the set of all values of \(x\) such that \(x\) is greater than 3."


Summary Checklist

1. Can you use inverse operations to solve for \(x\)?
2. Do you remember to do the same thing to both sides?
3. Can you expand brackets before solving?
4. Do you know the difference between an open \(\circ\) and closed \(\bullet\) circle on a number line?
5. (Higher Tier) Can you identify a region on a graph using a dashed or solid line?

Note: To learn about solving two equations at once, see the chapter on "Quadratic and simultaneous equations". To see how these equations look as lines, check "Linear graphs and coordinates".