Understanding Variables and Solving Inequalities

Think of variables as mystery boxes - they're letters that represent unknown numbers waiting to be discovered. When you combine these with numbers and operations, you create algebraic expressions that tell mathematical stories.

Understanding the vocabulary makes everything clearer. A coefficient is simply the number sitting next to your variable (like the 3 in 3x), whilst a term is any chunk of your expression separated by + or - signs. Constants are the reliable numbers that never change their value.

Inequalities work almost exactly like equations, but instead of an equals sign, you use symbols like > (greater than) or ≤ (less than or equal to). The golden rule? You can add, subtract, or multiply by positive numbers without changing the direction of the inequality sign.

Quick Tip: When you multiply or divide by a negative number, flip the inequality sign around - this catches loads of students out in exams!

Here's where it gets interesting: three-part inequalities like 2 < 2x < 6 can be solved by applying the same operation to all three parts. Divide everything by 2, and you get 1 < x < 3. Dead simple once you know the trick!

Mastering Equations and Formula Manipulation

Equations with variables on both sides might look intimidating, but there's a clever strategy: always move the variable with the smallest coefficient first. This keeps your working neat and reduces mistakes when you're solving under exam pressure.

Fractions in equations don't have to be your enemy. Find the least common denominator of all fractions, multiply everything by it, and watch those pesky fractions disappear completely. It's like magic, but with maths!

Changing the subject of a formula is essentially asking "what if I want to work out a different variable?" Use exactly the same techniques as solving equations - just rearrange until your chosen variable sits alone on one side.

Remember: When you square root both sides, don't forget the ± symbol - there are usually two possible answers!

Algebraic fractions follow the same rules as normal fractions, just with letters involved. To add them, find a common denominator. To multiply, go straight across (numerator × numerator, denominator × denominator). For division, flip the second fraction and multiply instead.

The key to mastering this topic? Practice recognising patterns and always work methodically through each step. Your future self will thank you when these skills become second nature!