Quadratic and Rational Inequalities

Transcript

0:00Introduction

Hello everyone! So, one quick question before we start. Solve this: two x minus three, over x plus one, is greater than or equal to one. Most students multiply both sides by x plus one. Two x minus three is greater than or equal to x plus one, so x is greater than or equal to four. Done, right? No, no! The correct answer also contains every x less than minus one. Half the answer, simply gone. Where did it go? Today we find out, with a method that never loses an answer. And look, JEE has asked inequalities like these many times. So, let's begin.

0:41Quadratic inequalities: find the roots first

So, first the simplest family: quadratic inequalities. We want a x squared, plus b x, plus c, greater than zero. Or less than zero, or with these signs that have a little line below: greater than or equal to, less than or equal to, okay? Step one: find the real roots. Call them alpha and beta, these two Greek letters, with alpha less than or equal to beta. Step two: use the sign rule from the start of this chapter. Clear? Now think. Suppose a is positive. When is x minus alpha, times x minus beta, less than zero? Between the roots, yes! There, x minus alpha is positive but x minus beta is negative, so the product is negative. So, less than zero if and only if x lies between alpha and beta: that is what this double arrow means. And for greater than zero, the opposite: outside the roots. x less than alpha, or x greater than beta. In short: between the roots, the opposite sign to a; outside, the same sign as a. That is our sign rule. That's it!

1:55Many factors: the wavy-curve method

Alright, but what if there are many factors, and even a denominator? Look at the figure. It is for x minus one, times x minus two, the whole squared, times x plus three, all over x minus four. The small two up there means x minus two is squared. And that red curve, waving above and below the number line, is why we call this the wavy-curve method, also called the sign scheme. Rule one: make every leading coefficient positive, so each factor starts with plus x. Here they already do. Then mark the critical points, where the top or the bottom becomes zero: minus three, one, two and four. Those are the dots on the number line. Rule two: start from the far right with plus, then move left. Why plus? For a very large x, every factor is positive, right? See the plus sign after four. So, move left, and cross four. Look, plus becomes minus. Why does the sign flip here? Hint: x minus four has power one. Yes, exactly! Multiplicity means the power on that factor, and here it is one: odd. Crossing four, only x minus four switches sign, so the whole expression flips. At a point of odd multiplicity, the sign changes. So between two and four: minus. Next stop, two. The figure says: double root, no sign change. But why? Look at the factor x minus two. It is squared! The power two is even, and a square never turns negative, so crossing two cannot flip the sign. Even multiplicity: the sign stays. So between one and two: still minus. Then crossing one, power one, odd: it flips to plus. Crossing minus three, odd again: back to minus. With me?

3:52Which critical points are included?

Last rule: pick the intervals you need. Zeros of the numerator make the expression zero, so they are included for greater than or equal to, or less than or equal to. Those are the filled dots: minus three, one and two. But zeros of the denominator are never included, because we can never divide by zero. That is the hollow dot at four, okay?

4:14The JEE trap from our hook

Now, the JEE trap, and the secret of our hook. Suppose x minus three sits in a denominator. Can we just multiply both sides by x minus three? No, never! Never multiply both sides of an inequality by an expression whose sign you do not know. Why? Because x minus three is negative for some x, and multiplying by a negative number flips the sign: greater than or equal to becomes less than or equal to. Exactly what happened in our hook, right? So, bring everything to one side, and use the sign scheme. Safe, every single time. Got it?

4:53Example 1: four quadratic inequalities, roots and D

Let's try four quick ones. The Roman numerals mark the parts. Part one: x squared minus five x plus six, less than zero. Part two: the same, greater than or equal to zero. Part three: x squared plus x plus one, greater than zero. Part four: x squared minus four x plus four, less than or equal to zero. Pause the video, and try it first. Done? Okay, let's check. In part one, it factorises as x minus two, times x minus three. Less than zero means between the roots: x between two and three. Part two is the same expression, but greater than or equal to zero. What changes? Outside the roots, yes! And now the roots themselves are allowed, because equal to zero is okay. So x less than or equal to two, or x greater than or equal to three. Part three. Here D, the discriminant, b squared minus four a c, is one minus four: negative. And a is one, positive. So the expression is always positive: all real x. Simple as that! Last one: x squared minus four x plus four is x minus two, the whole squared, and we want it less than or equal to zero. But a square is never negative, so there is no solution, right? Wait, careful! Read the sign again: less than or equal to. The square cannot be negative, but it can be zero, at x equals two. So the answer is only x equals two, one single point. Many students miss this one!

6:42Example 2: solve (2x - 3)/(x + 1) ≥ 1

And now, our hook question. Solve two x minus three, over x plus one, greater than or equal to one. Pause the video, and this time do it the safe way. Hint: no multiplying! Done? Okay, let's check. Take the one to the left side: two x minus three over x plus one, minus one. Same denominator, so the top becomes two x minus three, minus x, minus one, which is just x minus four. So we need x minus four, over x plus one, greater than or equal to zero. Neat, isn't it? Now your sign scheme. Where are the critical points, and what are the signs from the far right? The top is zero at four, the bottom at minus one. Both have power one, odd, so the sign changes at each: plus, minus, plus. Exactly! We want greater than or equal to zero, so the plus parts. But tell me: is four included? And minus one? Four, yes: it makes the numerator zero, and zero is allowed. Minus one, never: it makes the denominator zero. So, x less than minus one, or x greater than or equal to four. In intervals: minus infinity to minus one, union, four to infinity. The sideways eight is infinity, and the U shape means union. The round bracket leaves minus one out; the square bracket keeps four in. And our hook? The shortcut kept only the right part. The left part was lost because x plus one is negative there. Mystery solved!

8:23Quick revision: sign rule, wavy curve, which points count

So, quick revision. Quadratics: find the roots; outside, the sign of a; between, the opposite. Many factors: plus on the far right; odd powers change the sign, even powers keep it. And no multiplying by an unknown sign. For greater or equal, or less or equal, numerator zeros come in; denominator zeros, never. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!