$y \geq x^2+5$

What is the $x$-intercept?
What is the $y$-intercept?

To find the x -intercept, set y = 0 and solve for x . The inequality 0 ≥ x 2 + 5 has no real solutions, so there is no x -intercept.
To find the y -intercept, set x = 0 and solve for y . This gives y ≥ 0 2 + 5 , so y ≥ 5 .
The y -intercept is all y such that y ≥ 5 .
The x -intercept does not exist, and the y -intercept is y ≥ 5 , so the answer is y ≥ 5 ​ .

Explanation

Understanding the Problem We are given the inequality y g e x 2 + 5 . We need to find the x -intercept and y -intercept.

Finding the x-intercept To find the x -intercept, we set y = 0 and solve for x . This gives us 0 g e x 2 + 5 .

Analyzing the x-intercept inequality The inequality 0 g e x 2 + 5 can be rewritten as x 2 + 5 l e 0 . Since x 2 is always non-negative, x 2 g e 0 for all real numbers x . Therefore, x 2 + 5 g e 5 for all real numbers x . Thus, x 2 + 5 can never be less than or equal to 0. This means there is no x -intercept.

Finding the y-intercept To find the y -intercept, we set x = 0 and solve for y . This gives us y g e 0 2 + 5 , so y g e 5 .

Analyzing the y-intercept inequality The y -intercept is the set of all y such that y g e 5 . This means the y -intercept is [ 5 , ∞ ) .

Final Answer Therefore, there is no x -intercept, and the y -intercept is y g e 5 .


Examples
Understanding intercepts is crucial in various real-world scenarios. For instance, in economics, the y-intercept of a cost function represents the fixed costs a company must pay regardless of production levels. Similarly, in physics, the y-intercept of a velocity-time graph indicates the initial velocity of an object. Knowing how to find intercepts helps in analyzing and interpreting data in these and many other fields.

Answered by GinnyAnswer | 2025-07-05