Question

Solve x≤0 or x≥−4 and write the solution in interval notation.

Answer 1

Answer: $(-\infty, \infty)$

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Explanation:

Draw out a number line. Plot 0 and -4 on the number line.

Shade to the left of x = 0, and have a filled in circle at the endpoint. This is the graph of $x \le 0$

Then graph $x \ge -4$ by plotting a filled in circle at -4, and shading to the right.

Note how the two graphs overlap to cover the entire real number line

So if we have $x \le 0 \ \text{ or } \ x \ge -4$ then we're basically saying x is any real number. To write this in interval notation, we write $(-\infty, \infty)$

This is the interval from negative infinity to positive infinity (or just infinity). We exclude each endpoint because we can't actually reach infinity itself. Infinity is not a number. Infinity is a concept.

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Side note: if you change the "or" to "and", then the solution to $x \le 0 \ \text{ and } \ x \ge -4$ would be $[-4, 0]$ to indicate the interval from x = -4 to x = 0, including both endpoints. This is the region where the two graphs overlap.