Solving Inequalities: Finding the Greatest Integer Solution

Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving an inequality means finding the range of values that make the statement true. Sometimes, we need to find the greatest integer that satisfies the inequality. Let’s see how to do it!

The Problem:

Work out the greatest integer that satisfies the inequality:

5y – 7 < 16

Step 1: Solve the Inequality

Treat the inequality like an equation and isolate the variable ‘y’:

1. Add 7 to both sides:
   5y – 7 + 7 < 16 + 7
   5y < 23
2. Divide both sides by 5:
   5y / 5 < 23 / 5
   y < 4.6

Step 2: Identify the Greatest Integer

The inequality tells us that ‘y’ must be less than 4.6. We need to find the greatest integer (whole number) that is less than 4.6.

• The integers less than 4.6 are …, 2, 3, 4.

The greatest of these integers is 4.

Therefore, the greatest integer that satisfies the inequality 5y – 7 < 16 is 4.

In Summary

To solve an inequality and find the greatest (or smallest) integer solution:

1. Solve the inequality to isolate the variable.
2. Consider the integers that fall within the solution range.
3. Identify the greatest (or smallest) integer that satisfies the inequality.

Understanding inequalities is a vital skill for GCSE maths. If you’re finding it challenging, remember that help is available. A gcse maths tutor can provide support and guide you through these essential topics.