Final answer:
The solution set consists of the intervals (-∞, -4) U (5, +∞). This means that any real number less than -4 or greater than 5 will satisfy the inequality x² - x > 20.
Explanation:
To solve the inequality x² - x > 20, we can start by setting the expression equal to zero:
x² - x - 20 > 0
Next, we factorize the quadratic expression:
(x - 5)(x + 4) > 0
Now, we have two factors: (x - 5) and (x + 4). To determine the sign of the expression, we need to analyze the signs of each factor separately.
1. When (x - 5) > 0 and (x + 4) > 0:
- (x - 5) > 0 means x > 5
- (x + 4) > 0 means x > -4
In this case, both factors are positive. The solution set is the interval (5, +∞).
2. When (x - 5) < 0 and (x + 4) < 0:
- (x - 5) < 0 means x < 5
- (x + 4) < 0 means x < -4
In this case, both factors are negative. The solution set is the interval (-∞, -4).
3. When (x - 5) > 0 and (x + 4) < 0:
- (x - 5) > 0 means x > 5
- (x + 4) < 0 means x < -4
In this case, the first factor is positive and the second factor is negative. The solution set is the interval (-4, 5).
4. When (x - 5) < 0 and (x + 4) > 0:
- (x - 5) < 0 means x < 5
- (x + 4) > 0 means x > -4
In this case, the first factor is negative and the second factor is positive. The solution set is the union of two intervals: (-∞, -4) U (5, +∞).
U (5, +∞) and the intervals (-∞, -4) make up the solution set. This indicates that the inequality x² - x > 20 will be satisfied by any real number that is less than -4 or more than 5.