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Unformatted text preview: 11.4 Nonlinear Inequalities in One Variable Quadratic
lnegualities: A quadratic inequality is an inequality that can be written so that one
side is a quadratic expression and the other side is 0. Examples
3x2—4>0 2x2—5x+7SO
x2+4x—620 2x2—3<O A solution of a quadratic inequality is a value of the variable that makes
the inequality a true statement. Solving a Polynomial Inequality _1) Write the inequality in standard form, then solve the related
equation. ' 2) Separate the number line into regions with the solutions from
Step 1. 3) For each region, choose a test point and determine whether its
value satisfies the original inequality. 4) The solution set includes all the regions whose test point value
is a solution of the inequality.  If the inequality symbol is 2 or s, the values from Step 1
are solutions; ° If the inequalitysymbol is < or >, the values from Step 1
' are not included. If we attempt to solve a quadratic inequality, such as
3x2 + 5x + 2 < 0,
we are looking for values ofx that will make this a true statement. If we graph the quadratic equation y = 3x2 + 5x + 2, the points of the
parabola that lie below the x—axis would provide values ofx where the y—
value < 0. Hence, those values ofx would satisfy 3x2 + 5x + 2 < 0. Similarly, we could also use the graph to find the values ofx that satisfy
the inequality 3x2 + 5X + 2 > O (the xvalues of all points above the x—axis). The points on the graph above and below the x—axis are separated by
points actually on the xaxis. These points would have values of x such
that 3x2 + 5x+ 2 = 0. However, graphing a quadratic equation could be time consuming if you
don’t have a computer or graphing calculator. Fortunately, it is not necessary to graph a quadratic inequality to solve
this type of problem. We can construct a number line representing the xaxis and find the
region(s) on the number line where the inequality is true. Solve the quadratic inequality 3x2 + 5x + 2 < 0.
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 Fall '11
 JimCotter

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