Now that you know how to solve quadratic equations, you are ready to learn how to solve quadratic inequalities.
Quadratic inequalities
A
quadratic inequality is an inequality of the form
Solving a quadratic inequality corresponds to working out in what region the graph of a quadratic function lies above or below the
-axis.
Solve the inequality
and interpret the solution graphically.
Let
. Factorising this quadratic function gives
.
only when
.
This means that the graph of
touches the
-axis at
, but there are no regions where the graph is below the
-axis.
Find all the solutions to the inequality
.
The factors of
are
.
We need to figure out which values of
satisfy the inequality. From the answers we have five regions to consider.
Let
. For each region, choose any point in the region and evaluate the function.
sign of
Region A
+
Region B
+
Region C
-
Region D
+
Region E
+
We see that the function is positive for
and
.
We see that
is true for
and
.
Solve the quadratic inequality
.
Let
.
cannot be factorised so, use the quadratic formula to determine the roots of
. The
-intercepts are solutions to the quadratic equation
We need to figure out which values of
satisfy the inequality. From the answers we have five regions to consider.
We can use another method to determine the sign of the function over different regions, by drawing a rough sketch of the graph of the function. We know that the roots of the function correspond to the
-intercepts of the graph. Let
. We can see that this is a parabola with a maximum turning point that intersects the
-axis at
and
.
It is clear that
for
for
When working with an inequality where the variable is in the denominator, a different approach is needed.
Solve
We see that the expression is negative for
or
.
End of chapter exercises
Solve the following inequalities and show your answer on a number line.