Final Answer:
The solution set to the inequality x²-x-10 > 3x+11 is x < -2.5.
Step-by-step explanation:
To solve the inequality, we can follow these steps:
Move all terms to one side of the inequality:
x²-x-10 - (3x+11) > 0
x²-4x-21 > 0
Factor the expression:
(x+3)(x-7) > 0
Create a sign chart to analyze the inequality:
x x+3 x-7 (x+3)(x-7)
x < -7 - - +
-7 < x < -3 - + -
-3 < x < 7 + + +
x > 7 + + +
Interpret the sign chart:
The expression (x+3)(x-7) is positive when x < -7 or x > 7, and it is negative when -7 < x < -3. Since the inequality is > 0, we only want the values of x that make the expression positive. Therefore, the solution set to the inequality is x < -7 or x > 7.
Combine the two solutions:
Since we are looking for values that are either less than -7 or greater than 7, we can combine the two solutions into one: x < -7 or x > 7.
Express the solution in interval notation:
The solution set can be expressed in interval notation as x ∈ (-∞, -7) ∪ (7, ∞). This means that the solution set includes all real numbers less than -7 and all real numbers greater than 7.