Welcome to Linear Equations and Inequalities!

Have you ever tried to work out a mystery number, or wondered how much money you need to save to buy a game? If so, you have already been thinking algebraically! In this chapter, we will learn how to find unknown values using linear equations and describe ranges of numbers using inequalities.

Don't worry if algebra has felt confusing in the past. We will take everything one small step at a time, using simple analogies and clear patterns to make it feel natural.

1. Understanding Equations: The Balancing Scale

An equation is a mathematical statement showing that two expressions are equal. It always contains an equals sign, \(=\).

Think of an equation like a balanced set of scales. Whatever is on the left side weighs exactly the same as whatever is on the right side.

The Golden Rule of Equations:
Whatever operation you do to one side of the equation, you must do exactly the same to the other side to keep the scales balanced!

Inverse (Opposite) Operations

To find the mystery letter (usually \(x\)), we need to get it by itself on one side. We do this by "undoing" what has been done to it using 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 One-Step and Two-Step Equations

Example 1 (One-Step): Solve \(x + 7 = 15\)
To undo \(+ 7\), subtract \(7\) from both sides:
\(x + 7 - 7 = 15 - 7\)
\(x = 8\)

Example 2 (Two-Step): Solve \(3x - 4 = 11\)
Step 1: Deal with the constant term first. Undo \(- 4\) by adding \(4\) to both sides:
\(3x = 11 + 4\)
\(3x = 15\)
Step 2: Deal with the multiplying number (the coefficient). Undo \(\times 3\) by dividing both sides by \(3\):
\(x = \frac{15}{3}\)
\(x = 5\)

Key Takeaway for Basic Equations: Always work backwards! Usually, add or subtract loose numbers first, then multiply or divide to leave the letter completely on its own.

2. Equations with Brackets and Unknowns on Both Sides

Equations with Brackets

When an equation has brackets, the easiest first step is to expand (multiply out) the brackets before solving.

Example: Solve \(4(2x + 3) = 28\)
Step 1: Multiply everything inside the bracket by \(4\):
\(8x + 12 = 28\)
Step 2: Subtract \(12\) from both sides:
\(8x = 28 - 12\)
\(8x = 16\)
Step 3: Divide both sides by \(8\):
\(x = 2\)

Unknowns on Both Sides

Sometimes you will see \(x\) on both sides of the equals sign, such as \(5x + 2 = 2x + 14\).
Memory Trick: Eliminate the smaller \(x\) term first! This keeps your \(x\) terms positive and avoids dealing with negative signs.

Example: Solve \(5x + 2 = 2x + 14\)
Step 1: Compare \(5x\) and \(2x\). The smaller term is \(2x\). Subtract \(2x\) from both sides:
\(5x - 2x + 2 = 14\)
\(3x + 2 = 14\)
Step 2: Subtract \(2\) from both sides:
\(3x = 12\)
Step 3: Divide both sides by \(3\):
\(x = 4\)

Equations Involving Fractions

When an algebraic fraction appears, clear the denominator by multiplying every term by the bottom number.

Example: Solve \(\frac{x + 5}{3} = 4\)
Step 1: Multiply both sides by \(3\):
\(x + 5 = 4 \times 3\)
\(x + 5 = 12\)
Step 2: Subtract \(5\) from both sides:
\(x = 7\)

Common Mistake to Avoid: When expanding brackets like \(-3(x - 4)\), remember that multiplying two negative numbers gives a positive: \(-3 \times -4 = +12\), so \(-3(x - 4) = -3x + 12\).

Key Takeaway: Expand brackets first, eliminate the smaller variable term, and then solve as a simple two-step equation.

3. Forming and Solving Equations from Real-Life Situations

In your exam, you will often be given a word problem and asked to write an equation before solving it.

Step-by-Step Strategy:

1. Choose a letter (like \(x\)) to represent the unknown quantity.
2. Translate the words into mathematical expressions.
3. Set up the equation using the information given.
4. Solve the equation and check that your answer makes sense in the real world.

Example:
"A cinema ticket for an adult costs \(£4\) more than a child's ticket. A family buys \(2\) adult tickets and \(3\) child tickets. The total cost is \(£48\). Find the cost of a child's ticket."

• Let the cost of a child's ticket \(= x\)
• Then the cost of an adult ticket \(= x + 4\)
• Total cost: \(2(\text{adult}) + 3(\text{child}) = 48\)
• Form the equation: \(2(x + 4) + 3x = 48\)
• Expand brackets: \(2x + 8 + 3x = 48\)
• Combine like terms: \(5x + 8 = 48\)
• Subtract \(8\): \(5x = 40\)
• Divide by \(5\): \(x = 8\)
A child's ticket costs \(£8\).

Key Takeaway: Always define your variable clearly at the start (e.g., "Let \(x =\) ...").

4. Linear Inequalities

An equation tells us that two things are exactly equal. An inequality tells us that one quantity is larger or smaller than another.

Inequality Symbols

• \(<\) means strictly less than (e.g., \(x < 5\))
• \(>\) means strictly greater than (e.g., \(x > 2\))
• \(\le\) means less than or equal to (e.g., \(x \le 7\))
• \(\ge\) means greater than or equal to (e.g., \(x \ge -1\))

Memory Trick: Think of the inequality sign like an open crocodile mouth that always wants to eat the bigger value!

Representing Inequalities on a Number Line

We represent inequalities graphically using circles and arrows on a number line:

• Open Circle (\(\circ\)): Used for \(<\) and \(>\). It shows that the endpoint is not included.
• Closed/Solid Circle (\(\bullet\)): Used for \(\le\) and \(\ge\). It shows that the endpoint is included.
• Arrow: Points in the direction of all the possible solutions.

Example: Represent \(-1 \le x < 4\) on a number line.
• Place a solid circle at \(-1\) (because it has the \(\le\) symbol).
• Place an open circle at \(4\) (because it has the \(<\) symbol).
• Draw a line connecting the two circles.

Solving Linear Inequalities

Solving an inequality is almost identical to solving an equation: use inverse operations to isolate the variable!

Example: Solve \(4x - 5 \ge 11\)
Step 1: Add \(5\) to both sides:
\(4x \ge 16\)
Step 2: Divide both sides by \(4\):
\(x \ge 4\)

The Golden Rule for Inequalities (Danger Zone!)

Crucial Rule: If you multiply or divide both sides by a negative number, you must flip the inequality sign!

Why? Think about numbers: \(2 < 5\) is true. If we multiply both sides by \(-1\), we get \(-2\) and \(-5\). But \(-2\) is larger than \(-5\), so the sign must flip: \(-2 > -5\).

Example: Solve \(-3x < 12\)
Divide both sides by \(-3\) and flip the \(<\) to \(>\):
\(x > -4\)

Listing Integer Solutions

An integer is a whole number (positive, negative, or zero). Exam questions often ask you to list all integer values that satisfy an inequality.

Example: Write down all integers that satisfy \(-3 < x \le 2\).
• \(-3\) is NOT included (because of \(<\))
• \(2\) IS included (because of \(\le\))
The integer solutions are: \(-2, -1, 0, 1, 2\).

Key Takeaway for Inequalities: Treat inequalities like equations, but remember to use open/solid circles correctly on number lines, and always flip the inequality sign when dividing or multiplying by a negative number.

Quick Chapter Summary Checklist

Before you sit your exam, check that you can:
1. Solve basic two-step linear equations using inverse operations.
2. Expand brackets and collect like terms to solve equations with unknowns on both sides.
3. Write algebraic equations from real-life word descriptions.
4. Draw inequalities on a number line using open (\(\circ\)) and closed (\(\bullet\)) circles.
5. Solve inequalities and list integer solutions accurately.