Find the maximum value of a such that the following statement is true:
If ∣x−8∣<a, then x^2−11x−210<0.
To find the maximum value of "a" for which the given statement is true, we need to determine the range of values for "a" that satisfy the given condition.
First, let's analyze the given statement:
If ∣x−8∣<a, then x^2−11x−210<0.
This statement consists of two parts:
1. ∣x−8∣<a: This means that the distance between "x" and 8 is less than "a." In other words, x is within "a" distance (in either direction) from 8 on the number line.
2. x^2−11x−210<0: This is a quadratic inequality. To solve it, we need to find the values of "x" for which the quadratic expression is negative.
Let's start by solving the quadratic inequality. We want to find the critical points where the quadratic expression equals zero:
x^2−11x−210 = 0
Factoring the quadratic expression, we have:
(x−21)(x+10) = 0
Setting each factor equal to zero, we find two critical points:
x−21 = 0 => x = 21
x+10 = 0 => x = -10
Now let's analyze the quadratic expression:
x^2−11x−210
We can determine its sign by analyzing the intervals between and outside the critical points (-10 and 21):
For x < -10, the expression is positive because both factors (x−21) and (x+10) are negative.
For -10 < x < 21, the expression is negative because (x−21) is negative, and (x+10) is positive.
For x > 21, the expression is positive because both factors (x−21) and (x+10) are positive.
Now, let's relate this to the given condition: ∣x−8∣<a.
To make x^2−11x−210<0 true, we need the range of values for "x" between -10 and 21, where the expression is negative.
Therefore, the maximum value of "a" would be the distance between the critical points -10 and 21, which is:
a = |21 - (-10)| = 21 + 10 = 31.
So, the maximum value of "a" for which the statement is true is 31.