To make a sign chart, use the function and test values in each region bounded by the roots. Shade in the appropriate x-values depending on the original inequality. A sign chart gives us a visual reference that indicates where the function is above the x-axis using positive signs or below the x-axis using negative signs. We can streamline the process of solving quadratic inequalities by making use of a sign chart. If the inequality involves “greater than,” then determine the x-values where the function is above the x-axis. A quadratic inequality is of the form: a x 2 + b x + c > 0 Where a 0, and our > can be replaced with any inequality symbol. If the inequality involves “less than,” then determine the x-values where the function is below the x-axis. Graph the quadratic function and determine where it is above or below the x-axis. We can solve quadratic inequalities graphically by first rewriting the inequality in standard form, with zero on one side. Khan Academy is a 501(c)(3) nonprofit organization. Quadratic inequalities can have infinitely many solutions, one solution or no solution. Quadratic inequalities: graphical approach Our mission is to provide a free, world-class education to anyone, anywhere. It's a good idea to type them into a calculator to figure out what decimal they're approximately equal to, so that you can plot them on the number line as I have done in Figure 13.2. Unfortunately, this quadratic cannot be factored, so you'll have to use either the quadratic formula or complete the square to get the solutions, which will beīoy, those critical numbers sure are ugly. 
Solution: Pretend, for a moment, that this is actually the equation 2 x 2 + x - 2 = 0. They usually want you to express that interval as an inequality statement.Įxample 5: Solve the inequality and graph the solution. If the test value makes the inequality true, then so will all of the other values in that same interval, so it is part of the solution.Įven tough the graph you create during this process tells you what the solution is, most teachers would rather you not just give the graph as an answer. Choose a test value from each interval and plug it into the original inequality for x. The critical numbers will split the number line into segments, called intervals.
Use test values to find the solution interval(s). Just mark the critical numbers on the number line with either open or closed dots, depending upon whether or not the symbol allows for the possibility of equality (just like you did during the number line graphs of Linear Inequalities). 1) a) Complete the table and draw the graph of f(x) x2+3x+2. You're not changing that inequality sign to an equal sign permanentlyonce you find the critical numbers, you can change it back. Use in connection with the interactive file, Quadratic Inequalities on the Students CD. Those solutions are called the critical numbers of the inequality. Pretend the inequality symbol is an equal sign and solve the corresponding quadratic equation. They break the number line into segments, called intervals. The critical numbers are the values of x for which an inequality equals 0 or is undefined. 
That's enough talk for now let me explain the steps you'll follow to solve a quadratic inequality: For example, since 2 x 2 + x - 2 x (not x and y both), you should use a number line to graph it. The correct answer is a number line the number of axes in the graphing system should match the number of unique letters in the equation or inequality. What do you use to graph an inequality containing only one variablea number line or a coordinate plane? If you answered coordinate plane, oooh, I'm sorry, you're wrong, but at least you'll walk away with some fine parting gifts and a home version of our game. Remember back in Linear Inequalities when you learned that the solution to an inequality is often expressed as a graph, because there are almost always an infinite number of possible solutions? This is still true if the inequality contains a quadratic polynomial, like 2 x 2 + x - 2 In fact, there have been so many steps to follow in the last few topics, you probably feel like you're putting together Ikea furniture, but bear with me just a little longer and you'll know all you need to know about quadratics. So far, this section has been chock-full of procedures that require precise step-by-step instructions, and the final topic of the section will be no different. Solving One-Variable Quadratic Inequalities.
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