Math 2Name______
Notes Solving Quadratic Inequalities by Graphing
By now you have learned to solve quadratic equations using the quadratic formula or factoring. We have also discussed that solving the quadratic is the same as finding the roots or zeros of the equation. We are now going to investigate how to solve a quadratic inequality by graphing on the coordinate plane. If we apply what we know about the discriminant the only equations that can be graphed by hand are ones that have rational roots and therefore can be factored. So, we are going to use factoring and graphing to solve the quadratic inequalities.
Example 1:
Step 1: Solve as if the inequality equals 0.
Set = to 0:
Factor:
State the zeros:
Step 2: Graph both zeros on Step 3: Now that the zeros are plotted
the coordinate planeyoucan sketch the graph of the equation
Step 4: State the solutions to the inequality.
We need to look at where the graph is > 0. This means we are looking for all pieces of the graph above 0,where y = 0 or which is everything above the x-axis. If you were to highlight the parts of the graph that were above the x-axis, you would see that the solution would be and . You can not include -5 and -2 since it is > 0 therefore you do not underline the inequality sign.
Let’s say the inequality was:
Now you can include -5 and -2 so what’s the only difference in your solution? or
The inequality signs are underlined to show that -5
and -2 are included in the solution.
Example 2:
Step 1:
Multiply by -1 to make the leading coefficient positive
Remember to flip the inequality sign
Solve as if the inequality equals 0.
Factor
State the zeros
Step 2: Graph both zeros on Step 3: Now that the zeros are plotted
the coordinate planeyoucan sketch the graph of the equation
Step 4: State the solution to the inequality . To satisfy this inequality we are looking where the graph is , which is everything on or below the x-axis. In this case we can include the roots -3 and 1 in the solution. So the solution would be .
Let’s say the inequality was: , now the roots -3 and 1 can not be included so how does that change your answer? ______
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Example 3:
Step 1: Solve as if the inequality equals 0.
Set = to 0:
Factor:
State the zeros:
Step 2: Graph both zeros on Step 3: Now that the zeros are plotted
the coordinate planeyoucan sketch the graph of the equation
Step 4: State the solution to the inequality . To satisfy this inequality we are looking where the graph is , which is everything below the x-axis. In this case we can not include the root - ½ in the solution.
So the solution would be
Let’s say the inequality was: , now the root - ½ can be include so how does that change your answer? ______
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What is the solution if the inequality is:
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Example 4: Although the inequality is no longer 0 we can solve it the same as the examples above. The difference will be when we look at the graph to find the solution.
Step 1: Solve as if the inequality equals 0.
Set = to 0:
Factor:
State the zeros:
Step 2: Graph both zeros on Step 3: Now that the zeros are plotted
on the coordinate plane you can sketch
the graph of the equation
Step 4: State the solution to the inequality . To satisfy this inequality we are looking where the graph is , which is everything below the line y = 4. So rather than looking at everything above or below the x-axis where y = 0 we need to draw in a line where y = 4. So the solution would be .
Let’s say the inequality was: now to satisfy the inequality the solution is everything above the line y = 4.
State the final answer if this was the case:
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Math 2Name______
Worksheet Solving Quadratic Inequalities in One Variable
Solve the quadratic inequality:
Draw the graph and state the solution using algebraic and interval notation.
1. 2.
3. 4.