Solve the inequality and graph the solution.

n
2
≤4
Plot the endpoints. Select an endpoint to change it from closed to open. Select the middle of the segment, ray, or line to delete it.

1 answer

To solve the inequality \( n^2 \leq 4 \), we start by rewriting it in a more manageable form:

  1. Rearranging the inequality: \[ n^2 - 4 \leq 0 \]

  2. Factoring: This can be factored as: \[ (n - 2)(n + 2) \leq 0 \]

  3. Finding the critical points: The critical points occur when the expression is equal to zero: \[ n - 2 = 0 \quad \Rightarrow \quad n = 2 \] \[ n + 2 = 0 \quad \Rightarrow \quad n = -2 \] So the critical points are \( n = -2 \) and \( n = 2 \).

  4. Testing intervals: We will test intervals around the critical points to determine where the product \((n - 2)(n + 2)\) is less than or equal to zero:

    • Interval 1: \( n < -2 \) (e.g., \( n = -3 \)): \[ (-3 - 2)(-3 + 2) = (-5)(-1) = 5 \quad (\text{not } \leq 0) \]
    • Interval 2: \( -2 < n < 2 \) (e.g., \( n = 0 \)): \[ (0 - 2)(0 + 2) = (-2)(2) = -4 \quad (\text{ } \leq 0) \]
    • Interval 3: \( n > 2 \) (e.g., \( n = 3 \)): \[ (3 - 2)(3 + 2) = (1)(5) = 5 \quad (\text{not } \leq 0) \]
  5. Endpoints: The endpoints \( n = -2 \) and \( n = 2 \) are included in the inequality (because it is \( \leq 0 \)), so they are closed intervals.

Thus, the solution to \( n^2 \leq 4 \) is: \[ [-2, 2] \]

Graph of the Solution:

  • The graph will show a closed interval between \(-2\) and \(2\), which includes both endpoints.
  1. Plot the points \(-2\) and \(2\) and shade the region in between.
  2. Since we are specifically told to "select an endpoint to change it from closed to open," let's choose the endpoint \(2\) to make it open:

The final solution thus changes to: \[ [-2, 2) \]

Representation on a Number Line:

  • Mark \(-2\) with a closed dot (as it is included).
  • Mark \(2\) with an open dot (as it is not included).
  • Shade the region between \(-2\) and \(2\).

Here’s a textual representation:

<---|-----|-----|-----|-----|-----|-----|--->
    -3    -2    -1     0     1     2     3
           ●-------------◌

This visual representation shows that the solution set includes all numbers from \(-2\) to \(2\), excluding \(2\).