1. Word Problem worksheet questions

2. Quadratic Inequalities in One variable
What are the x-values where the inequality is true

3. Solving Quadratic Inequalities by Using Algebra (only one variable)
Solve the inequality x2 – 10x + 18 ≤ –3 by using algebra.
Step 1 Write the related equation.
x2 – 10x + 18 = –3

4. Step 2 Solve the equation for x to find the critical values.
Example Continued
Step 2 Solve the equation for x to find the critical values.
x2 –10x + 21 = 0
Write in standard form.
(x – 3)(x – 7) = 0
Factor.
x – 3 = 0 or x – 7 = 0
Zero Product Property.
x = 3 or x = 7
Solve for x.
The critical values are 3 and 7. The critical values divide the number line into three intervals: x ≤ 3, 3 ≤ x ≤ 7, x ≥ 7.

5. Step 3 Test an x-value in each interval.
Example 3 Continued
Step 3 Test an x-value in each interval.
–3 –2 –
Critical values
Test points
x2 – 10x + 18 ≤ –3
(2)2 – 10(2) + 18 ≤ –3
Try x = 2. x
(4)2 – 10(4) + 18 ≤ –3 
Try x = 4.
(8)2 – 10(8) + 18 ≤ –3 x
Try x = 8.

6. Example Continued
Shade the solution regions on the number line. Use solid circles for the critical values because the inequality contains them. The solution is {x|3 ≤ x ≤ 7} or [3, 7].
–3 –2 –

7. Another exampl When solving inequalities we are trying to
find all possible values of the variable
which will make the inequality true.
Consider the inequality
We are trying to find all the values of x for which the
quadratic is greater than zero or positive.

8. Solving a quadratic inequality
We can find the values where the quadratic equals zero
by solving the equation,
If it is not factorable, we can use the quadratic formula

9. Solving a quadratic inequality
For the quadratic inequality,
we found zeros 3 and –2 by solving the equation
. Put these values on a number line and we can see three intervals that we will test in the inequality. We will test one value from each interval. -2 3

10. Solving a quadratic inequality
Interval
Test Point
Evaluate in the inequality
True/False

11. Solving a quadratic inequality
You may recall the graph of a quadratic function is a parabola and the values we just found are the zeros or x-intercepts.
The graph of is

12. Solving a quadratic inequality
Thus the intervals is the solution set for the quadratic inequality,
In summary, one way to solve quadratic inequalities is to find the zeros and test a value from each of the intervals surrounding the zeros to determine which intervals make the inequality true.

13. Another way is to examine the parabola
Standard form ax²+bx+c = y
For a graph in ax²+bx+c≥0 form we have to find a solution where the y-value lies on or above the x-axis so we have to find its roots by using quadratic formula…

14. (For ≤, include the x-intercepts)
ax²+bx+c<0
→Graph the related quadratic function which is y=ax²+bx+c and find x-values for which the graph lies below the x-axis.
(For ≤, include the x-intercepts)
Shade outside
Shade inside
a<0
a>0
DON’T THINK ABOUT THE NUMBERS ON GRAPH, BUT JUST ITS SHAPE

15. ax²+bx+c>0
→ Graph the related quadratic function which is y=ax²+bx+c and find the x-values for which the graph lies above the x-axis. (For ≥, include the x-intercepts)
Shade outside
Shade inside
a<0
a>0
DON’T THINK ABOUT THE NUMBERS ON GRAPH, BUT JUST ITS SHAPE

16. Practice worksheet