To make solving multi-step linear inequalities easier, here are some clear steps you can follow:

1. Know the Inequality Symbols

Inequalities use special symbols, which are:

• $<$ (less than)
• $>$ (greater than)
• $\leq$ (less than or equal to)
• $\geq$ (greater than or equal to)

2. Treat Inequalities Like Equations

When solving inequalities, start by thinking of them as equations. Here are some basic steps:

1. Get the variable by itself: Try to move the variable to one side of the inequality.
2. Do the opposite operations: You can add, subtract, multiply, or divide, but make sure to keep the same direction for the inequality.

3. Important Rules for Multiplying/Dividing

When you multiply or divide by a negative number, you must flip the inequality sign. For example:

• If you have $-2x > 6$, dividing by $-2$ changes it to $x < -3$.

4. Making Complex Inequalities Simpler

For tricky inequalities, you can simplify them like this:

• Combine like terms: Add similar variables and numbers together.

  For example: If you solve $3x + 5 < 2x + 12$, first subtract $2x$ from both sides to get $x + 5 < 12$. Then, subtract 5 to find that $x < 7$.

• Distribute if you see parentheses: Use the distributive property. For instance, $2(3x - 4) \leq 8$ becomes $6x - 8 \leq 8$. Then, add 8 to both sides to get $6x \leq 16$, and finally divide by 6 to find $x \leq \frac{8}{3}$.

5. Use a Number Line

You can draw a number line to show your solutions. This helps you see what values work for the inequality:

• Open circles mean the number isn’t included (like $x < 2$).
• Closed circles show the number is included (like $x \leq 2$).

6. Practice to Improve

Try solving different inequalities to build your skills. Studies show that students who practice these problems do better, with over 70% earning a Grade 5 or higher on GCSE Maths tests by using these methods.

By following these simple steps, you can get better at solving multi-step linear inequalities!