![]() Is your brain throbbing with knowledge? That doesn't sound pleasant, but at least you're a bit smarter from the experience. For every inequality, the sign won't necessarily follow a predictable pattern from one root to the next. When working with polynomials larger than the quadratics there can be more than two roots, and we need to check the sign of every region. The expression is less than zero when -√2 < x < √2. It's even factored already, making things way easier. This inequality definitely isn't quadratic, but the method for finding the solutions is the same. So, is your brain starting to hurt? This next one is the last problem here, so stick with it a little longer. We want the regions that are greater than zero this time, so the solutions are: ![]() Now let's set up the number line and check the signs of each region. Our calculator tells us that these are x = -0.177 and x = -2.823. Now we need to use the quadratic formula to find our roots. Last time we had a nice, factorable equation to work with. ![]() We again start off by getting all our stuff on one side of the inequality, leaving a big fat zero on the other side. If you look back at the graph of the equation, you'll see that this is the region where it dips down below zero. Afterwards we'll put our cows in the positive regions, to boost cow morale.Īll the values of x between 1 and 4 will cause the inequality to be true. Those regions that are negative will be our solutions (since our inequality ends with "< 0"). We need to pick a point from each region to check whether it's positive or negative within that region. We'll now set up our roots on a number line, like so. Second, we can also use this technique to solve all kinds of polynomial inequalities, not just quadratic ones (see: the sample problem after that).Īnyway, back to solving. First, it will often be just as or more difficult to graph the equation than it will be to solve it the other way (see: the next sample problem). Hey, waitjustaminutehere! Couldn't we just graph the equation and solve it visually? We could, but there are two good reasons not to. We'll use this to help us find our solutions. The only places that it can possibly change sign (from above zero to below, or vice versa) are at the roots. So what? Take a look at the graph of this equation.Ī parabola is a smooth, continuous curve. Set the equation "equal" to zero, and then solve to find the roots of the equation. It's a good thing that the first step of solving an inequality is to pretend that the inequality is an equal sign. You can then fence off those regions and raise cows on them. So our answers won't be single values, but large, sweeping regions of number space. When we solve an inequality, what we want are all the values of x that make the statement true. We chose x = 0 and evalute the left side of the inequality.Remember back when we looked at linear inequalities? At one point, we said, "We promise we'll try to make your brain hurt more later." We always keep our promises, so grab an ice pack and strap in, because now we're going to look at quadratic inequalities.The two associated two-variable equations in this case are y 2 x2 + 4 x and y x2 x 6. What we need to do is to find this sign using one test value only Solving Quadratic Inequalities: Examples (page 2 of 3) Solve 2x2 + 4x > x2 x 6. Since the discriminant is negative, the left side x 2 + x + 4 of the inequality has no zeros and therefore has the same sign over the interval (- ∞, + ∞).We need values of x for which (3x + 4)(x - 1) is greater than 0, hence the solution set.chose x = 4 and evaluate (3x + 4)(x - 1).chose x = 0 and evaluate (3x + 4)(x - 1).chose x = - 2 and find the sign of (3x + 4)(x - 1).We chose a real number within each interval and use.The two real zeros - 4 / 3 and 1 of the left side of the inequality, divide the real number line into 3 intervals.Factor the left side of the inequality.Since the discriminant is positive, the left side 3 x 2 + x - 4 of the inequality has two zeros at which the sign changes. ![]() Rewrite the inequality with one side equal.
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