Question

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

Answer 1

The solution to the inequality x≤0 or x≥−4 is expressed in interval notation as (-∞, 0] ∪ [-4, +∞), representing all real numbers less than or equal to 0 or greater than or equal to -4.

Step-by-step explanation:

To solve the inequality x≤0 or x≥−4, we need to consider each part separately. The first inequality, x≤0, means that x can be any number that is less than or equal to 0. The second inequality, x≥−4, means that x can be any number that is greater than or equal to -4. The union of these two sets covers all real numbers from negative infinity up to and including zero as well as all real numbers from -4 to positive infinity.

The solution in interval notation is (-∞, 0] ∪ [-4, +∞), which represents all numbers less than or equal to 0 or greater than or equal to -4.

Answer 2

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

=================================================

Step-by-step 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.

-------------

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.