if the domain is the set of integers, the solution set of x^2 -3x -4<0 is what?
What your question really asks for is,
"For what values of x does the corresponding parabola
y = x^2 - 3x - 4 fall below the x-axis?"
let's find the x-intercepts ....
x^2 - 3x - 4 = 0
(x-4)(x+1) = 0
x = -1 or x = 4
so the expression is < 0 for -1 < x < 4
or
(x+1)(x-4) < 0
( x+1 >0 AND x-4<0) OR ( x+1 < 0 AND x-4>0)
( x>-1 and x<4) OR (x<-1 AND x > 4)
x>-1 and x<4 OR (null set)
-1 < x < 4
thank you, so anything between -1 and 4 would be my answer? (0,1,2,3)?
yes.
Since your domain is integers, only the integers (0,1,2,3) satisfy the inequality. Otherwise, there would be an infinite number of non-integer solutions between -1 and 4.
To find the solution set of the inequality x^2 - 3x - 4 < 0 when the domain is the set of integers, we need to solve the inequality algebraically. Here's how we can do it step by step:
1. First, let's factorize the quadratic equation x^2 - 3x - 4. We need to find two numbers that multiply to give -4 and add up to -3. The factors of -4 are (-1, 4) and (-2, 2). The pair (-4, 1) won't work as their sum is -3. Therefore, we can factor the equation as (x - 4)(x + 1).
2. Next, we set each factor equal to zero and solve for x to find the critical points:
x - 4 = 0 --> x = 4
x + 1 = 0 --> x = -1
3. Now, we have three intervals on the number line: (-∞, -1), (-1, 4), and (4, ∞). We need to determine which intervals satisfy the inequality x^2 - 3x - 4 < 0.
4. To do this, we can use test points within each interval and see if they satisfy the inequality. For example, let's use -2 as a test point in the interval (-∞, -1):
(-2)^2 - 3(-2) - 4 = 4 + 6 - 4 = 6 > 0
Since the test point (-2) does not satisfy the inequality, this interval is not part of the solution set.
5. Let's test a point within the interval (-1, 4). We can use 0 as our test point:
0^2 - 3(0) - 4 = -4 < 0
Since the test point 0 satisfies the inequality, this interval (-1, 4) is part of the solution set.
6. Lastly, let's test a point within the interval (4, ∞). We can use 5 as our test point:
5^2 - 3(5) - 4 = 25 - 15 - 4 = 6 > 0
Since the test point 5 does not satisfy the inequality, this interval is not part of the solution set.
Therefore, the solution set, when the domain is the set of integers, is (-1, 4).