2. One main difference between quadratic and linear inequalities is that at the moment of graphing, the linear inequality will be represented as a line, while the quadratic inequality can be represented as a parabola. This characteristic is done by the x of each type of equation, in the linear equations x is not raised to any power, however in the quadratic equations x is raised to the second power.

3. 1) X-intercepts: find the x-intercepts, which are the “=0” points. You can find them by factorization, quadratic formula or completing the square. Graph the “=0” points and check if the parabola will be negative or positive. You can know this by checking if the “x²” is positive or negative.

2) Intervals: divide the number line into 3 intervals. You’ll get (-∞, a) (a,b) (b, ∞).

3) Test values: choose a number that is between the 2 x-coordinates that you found and substitute that number in the original quadratic inequality and solve it. If the result is greater than 0, the parabola will be above the x-intercept. If the result is less than 0, the parabola will be below the x-intercept.

4) Determine the intervals: Express the intervals that fullfil the equation.

4. (Example is in photos at the end.) First photo: Steps 1 to 3

Second photo: Step 4

References

Angel A. R., Runde D. C. (2011) Quadratic and other inequalities in one variable. Intermediate Algebra for College Students. U.S.A.: Pearson Education. p. 548

Virtual Nerd (2015) What is a quadratic inequality. Virtual Nerd. Retrieved from: http://www.virtualnerd.com

a. Define a Quadratic Inequality 

Any inequality relationship that involves a quadratic function is called a quadratic inequality.

For example: (x - 2)(2x + 1) > 0

b. What is the difference between a linear inequality and a quadratic inequality?

The difference between a linear inequality and a quadratic inequality is that at the moment to graph a quadratic inequality the graph is shown as a parabola, while at graphing the linear inequality the graph is a straight line.

c. Determine how to solve a quadratic inequality with steps

1. Use x-Intercepts: Solve the inequality as though it were an equation. Solve for x.

2. Intervals: Make the boundary points solid circles if the original inequality includes equality; otherwise, make the boundary points open circles.

3. Test Values: Select points from each of the regions created by the boundary points. Replace these “test points” in the original inequality.

4. Determine the intervals that fulfills the inequality conditions: If a test point satisfies the original inequality, then the region that contains that test point is part of the solution.

d. Do an example with all the previous steps (c) showing the graph to prove your answer

Example: x² -x-6<0

Factorize

x² -x-6<0

(x-3)(x+2)<0

Equal to zero

(x-3)(x+2)=0

x=3, x=-2

Between the -2 and 3 the function will be either greater than zero or less than zero

We pick a value

x=0

(0)²-0-6<0

-6<0

So between the -2 and 3 the function is less than zero

In Interval would be (-2,3)

References:

Bostock, L., & Chandler, S. (1995). Quadratic Functions: Quadratic Inequalities. In Pure Mathematics 1. London: Stanley Thornes.

N.a.(2014) Cliffsnotes, March 24, 2015. Retrieved from  http://www.cliffsnotes.com/math/algebra/algebra-ii/quadratics-in-one-variable/solving-quadratic-inequalities

2. A quadratic inequality is graphed as a parabola while the linear inequality is graphed as a line

3. Step 1. Transform the inequality into a trinomial f(x) on the left side and leave 0 on the right side.

Step 2. Solve the quadratic equation to get the real roots.

Step 3. Solve the quadratic inequality, based on the values of the two real roots.

Step 4. Express the answer (or solution-set) in the form of intervals.

http://encyclopedia2.thefreedictionary.com/quadratic+inequality

http://www.wikihow.com/Solve-Quadratic-Inequalities

http://www.regentsprep.org/regents/math/algtrig/ate6/quadinequal.htm