Proving a Simple Inequality

In the world of mathematics, inequalities are just as important as equations. They help us define ranges and boundaries, and proving them can be a fun exercise in algebraic manipulation. Let’s tackle a straightforward example:

The Inequality:

Prove that x^2 + x + 2 > 1 for all real values of x

The Proof:

Rearrange the Inequality:

First, we want to get everything on one side and compare it to zero. Subtract 1 from both sides: x^2 + x + 1 > 0

Complete the Square:

Completing the square is a powerful technique that allows us to rewrite quadratic expressions in a more insightful form. Take the coefficient of our  term (which is 1), divide it by 2 (giving us 1/2), and square it (giving us 1/4). Add and subtract this value within the expression:

x^2 + x + \frac{1}{4} - \frac{1}{4} + 1 > 0

Now, we can rewrite the first three terms as a perfect square:

(x + \frac{1}{2})^2 + \frac{3}{4} > 0

Since a non-negative number plus a positive number is always positive, we can confidently say that: (x + \frac{1}{2})^2 + \frac{3}{4} > 0 or all real values of x.

Conclusion:

We have successfully proven that x^2 + x + 2 > 1 for all real numbers x by completing the square and analyzing the resulting expression. This demonstrates a common strategy in inequality proofs: manipulate the expression into a form where its sign is easily determined.

If you’re finding these types of problems challenging and need additional support to master these techniques, consider reaching out to an A Level Online Maths Tutor. They can provide tailored guidance and help you excel in your math studies.