Pick a point not on the parabola and see if it makes a true statement.
Either the inside or the outside of the parabola is shaded.If the equation is > or What is different in the graphing process of an equality and an inequality? Test an x-value in each interval to see if it satisfies the inequality. The critical x-values partition the number line into three intervals. Plot – 3 and 5 on a number line, using open dots because the values do not satisfy the inequality. The numbers – 3 and 5 are the critical x-values of the inequality x2 – 2x > 15. x2 – 2x = 15 Write equation that corresponds to original inequality. SOLUTION First, write and solve the equation obtained by replacing > with =. The graph lies above the x-axis when 4.2 15 algebraically. For what years was the number of teams greater than 100? You want to find the values of xfor which: T(x) > 100 7.51x2 – 16.4x – 65 > 0 Graph y = 7.51x2 – 16.4x – 65 on the domain 0 ≤ x ≤ 9. Robotics The number Tof teams that have participated in a robot-building competition for high school students can be modeled by T(x) = 7.51x2 –16.4x + 35.0, 0 ≤ x ≤ 9 Where x is the number of years since 1992. STEP 3 Identify the purple regionwhere the two graphs overlap. The graph is the blue region inside (but not including) the parabola y = x2 –2x – 3.
The graph is the red region inside and including the parabola y = –x2 + 4. Solving a Quadratic Inequality by Graphing (Looking for x intercepts) x2 − 6x + 5 x2 – 2x – 3 Inequality 2 SOLUTION STEP 1 Graph y ≤ –x2 + 4. Test a point inside the parabola, such as (1, 2000).Ģ000 ≤ 1480 Rappelling continued W≤ 1480d 2 2000≤ 1480(1)2 Because (1, 2000) is not a solution, shade the region below the parabola. Because the inequality symbol is ≤, make the parabola solid. SOLUTION Graph W = 1480d 2 for nonnegative values of d. Rappelling A manila rope used for rappelling down a cliff can safely support a weight W(in pounds) provided W ≤ 1480d 2 where dis the rope’s diameter (in inches). If you get a false statement when you substitute the point into the inequality, shade the opposite area.If you get a true statement when you substitute the point into the inequality, shade that area.Test a point inside the parabola, such as (0, 0).Y > ax2 + bx + c y ≥ ax2 + bx + c y x2 − 2x − 3 x = −b/2a= y = Look at the following examples, where we apply this process to solve quadratic inequalities.Graphing & Solving Quadratic Inequalities 5.7 What is different in the graphing process of an equality and an inequality? How can you check the x-intercepts of a quadratic equation or inequality? Why is it important to test 3 intervals after you have found the critical x values? We can accomplish this by using the graph or the statements from step 3. Step 4: Determine the inequality symbols that will make the solutions found in step 2 satisfy the inequality. Values above the x-axis are greater than 0, and values below the x-axis are less than 0.When the quadratic term is negative, the parabola opens down.When the quadratic term is positive, the parabola opens up and is U-shaped.Step 3: Obtain a simple graph of the function $latex y=ax^2+bx+c$, or alternatively, consider the following: We can solve this by factoring the quadratic expression. These are the points where the function $latex y=ax^2+bx+c$ intersects the x-axis.
Step 2: Find the values of x such that $latex ax^2+bx+c=0$. Note: The “>” sign will depend on the problem. Step 1: If it isn’t, write the inequality in the form $latex ax^2+bx+c>0$. We can solve quadratic inequalities by following these steps: The following are examples of quadratic inequalities: Quadratic inequalities are quadratic expressions that use inequality signs to compare two quantities.
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