MathematicsQuadratic Inequalitiesarkiteklea@gmail.com

Quadratic Inequalities Example:   Solve using a sign graph of factors, write your answer in interval notation and graph the solution

set: 

Step 1: Write the quadratic inequality in standard form. The given quadratic inequality is already in standard form.

Step 2: Solve the quadratic equation, , by factoring to get the boundary point(s).

-5 and 3 are boundary points.

Step 3: Use the boundary points found in Step 2 to mark off test intervals on the number line and list all of the factors found in Step 2.

Below is a graph that marks off the boundary points -5 and 3 and shows the three sections that those points have created on the graph.  Note that open holes were used on those two points since our original inequality did not include where it is equal to 0.   

Note that the two boundary points create three sections on the graph:  ,  , and  .

Step 4: Find the sign of every factor in every interval.  You can choose ANY point in an interval to represent that interval.  Remember that we are not interested in the actual value that we get, but what SIGN (positive or negative) that we get.

If we chose a number in the first interval, , like -6 (I could have used -10, -25, or -10000 as long as it is in the interval), it would make both factors negative: -6 + 5 = -1 and -6 - 3 = -9

“Tell me and I'll forget; show me and I may remember; involve me and I'll understand.” –Chinese Proverbs

MathematicsQuadratic Inequalitiesarkiteklea@gmail.com

If we chose a number in the second interval,  , like 0 (I could have used -4, -1, or 2 as long as it is in the interval), it would make x + 5 positive and x - 3 negative:  0 + 5 = 5 and 0 - 3 = -3

If we chose a number in the third interval, , like 4 (I could have used 10, 25, or 10000 as long as it is in the interval), it would make both factors positive:  4 + 5 = 9 and 4 - 3 = 1

Step 5: Using the signs found in Step 4, determine the sign of the overall quadratic function in each interval.

In the first interval, , we have a negative times a negative, so the sign of the quadratic in that interval is positive.

In the second interval, , we have a positive times a negative, so the sign of the quadratic in that interval is negative.

In the third interval,  , we have two positives, so the sign of the quadratic in that interval is positive.

Keep in mind that our original problem is .  Since we are looking for the quadratic expression to be LESS THAN 0, that means we need our sign to be NEGATIVE.

It looks like the only interval that this quadratic is negative is the second interval, .

Step 6: Write the solution set and graph.

*Open interval indicating all values between -5 and 3 *Visual showing all numbers between -5 and 3 on the number line

 

“Tell me and I'll forget; show me and I may remember; involve me and I'll understand.” –Chinese Proverbs