Solve the quadratic inequality. Enter your answer in interval notation. Use inf for [tex]$\infty$[/tex].
[tex]$x^2-x \geq 6$[/tex]
Answer (1)
Rewrite the inequality: x 2 − x − 6 ≥ 0 .
Factor the quadratic: ( x − 3 ) ( x + 2 ) ≥ 0 .
Find critical points: x = − 2 and x = 3 .
Test intervals and include critical points: The solution is ( − ∞ , − 2 ] ∪ [ 3 , ∞ ) .
Explanation
Understanding the Problem We are given the quadratic inequality x 2 − x ≥ 6 . Our goal is to solve for x and express the solution in interval notation, using 'inf' for ∞ .
Rewriting the Inequality First, we rewrite the inequality so that one side is zero. Subtracting 6 from both sides, we get x 2 − x − 6 ≥ 0
Factoring the Quadratic Next, we factor the quadratic expression x 2 − x − 6 . We are looking for two numbers that multiply to -6 and add to -1. These numbers are -3 and 2. Thus, we can factor the quadratic as ( x − 3 ) ( x + 2 ) ≥ 0
Finding Critical Points To find the critical points, we set each factor equal to zero and solve for x :
x − 3 = 0 ⇒ x = 3 x + 2 = 0 ⇒ x = − 2
Testing Intervals Now, we create a number line and test the intervals determined by the critical points: ( − ∞ , − 2 ) , ( − 2 , 3 ) , and ( 3 , ∞ ) .
For the interval ( − ∞ , − 2 ) , let's choose a test point x = − 3 . Plugging this into the factored inequality, we get: ( − 3 − 3 ) ( − 3 + 2 ) = ( − 6 ) ( − 1 ) = 6 ≥ 0 Since this is true, the interval ( − ∞ , − 2 ) is part of the solution.
For the interval ( − 2 , 3 ) , let's choose a test point x = 0 . Plugging this into the factored inequality, we get: ( 0 − 3 ) ( 0 + 2 ) = ( − 3 ) ( 2 ) = − 6 ≥ 0 Since this is false, the interval ( − 2 , 3 ) is not part of the solution.
For the interval ( 3 , ∞ ) , let's choose a test point x = 4 . Plugging this into the factored inequality, we get: ( 4 − 3 ) ( 4 + 2 ) = ( 1 ) ( 6 ) = 6 ≥ 0 Since this is true, the interval ( 3 , ∞ ) is part of the solution.
Including Critical Points Since the inequality is non-strict ( ≥ ), we include the critical points in the solution. Therefore, the solution is ( − ∞ , − 2 ] ∪ [ 3 , ∞ ) .
Final Solution Expressing the solution in interval notation using 'inf' for ∞ , we have ( − inf , − 2 ] ∪ [ 3 , inf ) .
Examples
Quadratic inequalities are useful in various real-world scenarios. For instance, a company might use them to model the profit margin based on the number of units sold. If the profit is described by a quadratic function, solving a quadratic inequality can help determine the range of units that need to be sold to achieve a certain profit level. Similarly, in physics, quadratic inequalities can be used to describe the range of projectile motion, such as determining the launch angles that allow a ball to reach a certain height or distance. These applications highlight the practical importance of understanding and solving quadratic inequalities.
