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Inequalities – Finding when a line in below a parabola – Exercise 5719

Exercise

Find the section in x-aixs where the line:

y=x+1

is below the parabla:

y=x^2-2x-9

Final Answer


x<-2 \text{  or  } x>5

Solution

We want to find for which points in x-aixs the following inequality is true:

x^2-2x-9>x+1

Move everything to one side:

x^2-2x-9-x-1>0

x^2-3x-10>0

It is a square inequality. Its coefficients are

a=1, b=-3, c=-10

The coefficient of the square expression (a) is positive, so the parabola (quadratic equation graph) “smiles” (= bowl-shaped). The sign of the inequality means we are looking for the sections the parabola is above the x-axis. We find the solutions (= zeros = roots) of the quadratic equation using the quadratic formula. Putting the coefficients in the formula gives us

x_{1,2}=\frac{3\pm \sqrt{{(-3)}^2-4\cdot 1\cdot (-10)}}{2\cdot 1}=

=\frac{3\pm \sqrt{49}}{2}=

=\frac{3\pm 7}{2}

Hence, we get the solutions:

x_1=\frac{3+ 7}{2}=5

x_2=\frac{3- 7}{2}=-2

Because the parabola we got “smiles” and we look for the section where it is above the x-axis, we get that the requested section is

x<-2 \text{  or  } x>5

Here are the line and the parabola:

פרבולה וישר

You can see that the line is indeed below the parabola in the section we found.

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