how do you solve an inequality? x^2-x-20<0please help!!

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to solve the attached equation,`x^2 - x - 20 = 0` , such that:

`x^2 - x - 20 = 0`

You may use quadratic formula to find the solutions `x_1`  and `x_2 ` such that:

`x_(1,2) = (-b +- sqrt(b^2 - 4ac))/(2a)`

You need to identify the coefficients a,b,c such that:

`a = 1, b = -1, c = -20`

`x_(1,2) = (1 +- sqrt(1 - 4*1*(-20)))/(2*1)`

`x_(1,2) = (1 +- sqrt 81)/2 => x_(1,2) = (1 +- 9)/2`

`x_1 = 5 ; x_2 = -4`

You need to draw a number line and you need to plot the solutions to the number line splitting the number line in three intervals, `(-oo,-4), (-4,5), (5,oo).`

Notice that the solutions `x_1 = 5 ; x_2 = -4`  are not included, since the quadratic expression is stricly negative.

You need to pick a number in each interval and test it into the given inequality to check if the inequality holds.

You need to pick `-5 in (-oo,-4)`  and you need to substitute -5 for x in `x^2 - x - 20 < 0`  such that:

`(-5)^2 + 5 - 20 < 0 => 25 + 5 - 20 < 0 => 10 < 0`  invalid

You need to pick `0 in (-4,5)`  and you need to substitute 0 for x in `x^2 - x - 20 < 0`  such that:

`0^2 - 0 - 20 < 0 => -20 < 0`  valid

You need to pick `6 in (5,oo) ` and you need to substitute 6 for x in `x^2 - x - 20 < 0`  such that:

`36 - 6 - 20 < 0 => 10 < 0`  invalid

Notice that the quadratic is negative where the graph is below x axis, hence `x in (-4,5).`

Hence, evaluating the solution to the given inequality yields the interval `(-4,5).`

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