Question

Tell whether you should use an open dot or a closed dot on the graph of the inequality. \(-2 x-1 \leq 3\)

Short answer

A closed dot will be used to represent the solution on the graph of the inequality.

Step-by-step solution

Solve for 'x'

First, solve the given inequality for 'x'. Start by adding '2x' to both sides of the inequality: \[-2x + 2x - 1 \leq 3 + 2x.\] This simplifies to \[-1 \leq 3 + 2x.\] Then, subtract '3' from both sides of the inequality: \[-1 - 3 \leq 3 - 3 + 2x.\] Simplifying it gives us \[-4 \leq 2x.\] Finally, divide every term by '2' to solve for 'x': \[-4/2 \leq 2x/2.\] So, \(x \geq -2\).

Determining the type of dot to use

For the inequality \(x \geq -2\), the 'greater than or equal to' relation suggests that '-2' is included in the set of solutions. So, to represent this on a number line, a closed dot should be used.