Solving Inequalities Graphically
Dr. Howard
Steps in solving an inequality graphically using Scientific Notebook:
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Re-arrange the inequality so that an expression in x (refer to that
expression as " f(x)" ) is on one side of the inequality and zero
is on the other side.
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Draw the graph of y = f(x). If necessary, adjust the
graph window to include all x values determined in step 2.
To adjust the graph window:
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Position the mouse over the graph, then right-click to select the graph.
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Click on the small colored box in the lower right-hand corner of the graph
window.
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Choose the ''Plot Components'' tab.
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Under the ''Domain Intervals'' heading, change the smaller and larger x
values so that all solutions found in step 2 fit within the selected interval.
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Choose the ''View'' tab.
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Under the ''View Intervals'' heading, un-check the ''Default'' box if it
is checked. Adjust the y interval as appropriate. It may require
some trial and error to determine a suitable view of the graph.
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Solve the inequality:
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If you need to solve f(x) < 0, determine the
x
values for which the graph of y = f(x) lies below the x-axis.
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If you need to solve f(x) > 0, determine the x
values for which the graph of y = f(x) lies above the x-axis.
Example
For example, suppose we wish to solve the inequality
2x2 + 5x > 12. We first re-write the inequality
as 2x2 + 5x - 12 > 0, so that we have f(x)
= 2x2 + 5x - 12. Then examine the graph
of y = 2x2 + 5x - 12 to see where
it lies the x-axis (since we wish to
determine x values for which y
0 ). Here is the graph:
The blue shading indicates the x-values for which the graph lies
above the x-axis. From the graph, we deduce that the solution
of the inequality is the pair of intervals ( - infinity, -4 ) and ( 3/2,
infinity ).
Assignment:
Use the steps outlined above to solve the following inequalities.
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x3 - 4x2 > 11x - 30
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6x3 - 7x2 + 15 < 14x
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x2 + 1 > 0
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(x+1)2 < 0
Here is what you must turn in:
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A sheet (may be hand-written or may be typed using Scientific Notebook)
showing each original inequality and the solution to the inequality written
in interval form.
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A printout showing your graph in Scientific Notebook. Clearly indicate
the problem number before the work for each exercise. Shade in the
portion of the graph corresponding to the solution of the inequality (like
the blue shading in the graph above).