Welcome to inequalities! Unlike equations which have exactly one solution, inequalities represent ranges of values. For example, x equals 5 has only one answer, but x is greater than 5 includes all numbers above 5. We show this on number lines - equations use solid dots for single solutions, while inequalities use arrows to show multiple solutions.
There are four main inequality symbols to master. Greater than uses the symbol that opens to the right, less than opens to the left. We also have greater than or equal to, and less than or equal to, which include the boundary value. Remember the crocodile mouth memory trick - the mouth always opens toward the bigger number! So 5 is less than 8, and 9 is greater than 3.
Now let's solve simple inequalities using addition and subtraction. For x plus 3 greater than 7, we subtract 3 from both sides, just like with equations. This gives us x greater than 4. On the number line, we show this with an open circle at 4 and an arrow pointing right. For x minus 2 less than or equal to 5, we add 2 to both sides, getting x less than or equal to 7. We show this with a solid dot at 7 and an arrow pointing left.
When solving inequalities with multiplication and division, there's a crucial rule to remember. For positive numbers, like 2x greater than 6, we divide both sides by 2 and keep the same inequality sign, giving us x greater than 3. However, when we multiply or divide by a negative number, we must flip the inequality sign! For negative 2x greater than 6, dividing by negative 2 flips the sign, so we get x less than negative 3. This is because multiplying by negative numbers reverses the order of values.
Now let's solve multi-step inequalities by combining all our skills. For 3x plus 5 less than or equal to 14, we first subtract 5 from both sides, getting 3x less than or equal to 9. Then divide by 3 to get x less than or equal to 3. For negative 2x plus 7 greater than 1, we subtract 7 from both sides, getting negative 2x greater than negative 6. When we divide by negative 2, we flip the sign, giving us x less than 3. Always remember to flip the inequality when dividing by negative numbers!