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For the graphic method, start by graphing the associated quadratic function. For the analytic method, start by replacing the inequality symbol with an equal sign, then find the roots for the equation obtained.
See solution.
Let's start by graphing the corresponding quadratic function. This will allow us to identify the values for which the quadratic inequality holds true.
For this inequality we are just interested in the x-values for which the associated function f(x)=x2+6x−8 takes negative values. Let's identify them in the graph.
As we can see, the function goes below the x-axis approximately at x=-7.1. Then, it starts going above it at approximately x=1.1. Therefore, we can approximate the solution of the inequality set as −7.1<x<1.1.
x=-3±17 | |
---|---|
x=-3+17 | x=-3−17 |
x=-3+17 | x=-3−17 |
x=-3+4.123105… | x=-3−4.123105… |
x=1.23105… | x=-7.123105… |
x≈1.123 | x≈-7.123 |
Now we know the critical values of the inequality. We can plot them on a number line using open points. This is because these values do not satisfy the inequality, since the inequality symbol is <. Note that the critical values partition the number line into three intervals.
We can use a test value from each of these intervals to determine whether they satisfy the inequality or not.
1st Interval | 2nd Interval | 3rd Interval | |
---|---|---|---|
Test Value | x=-8 | x=0 | x=2 |
Substitute in x2+6x−8<0 | (-8)2+6(-8)−8<?0 | (0)2+6(0)−8<?0 | (2)2+6(2)−8<?0 |
Is it a solution? | 10≮0 × | -8<0 ✓ | 8≮0 × |
Now that we know which intervals belong to the solution set and which of them do not, we can graph the inequality.