Question

Write a compound inequality for the graph shown below.

F $\mathbf{-}\mathbf{2}\mathbf{\le }\mathit{x}\mathbf{<}\mathbf{3}$

G $\mathit{x}\mathbf{\le }\mathbf{-}\mathbf{2}$or $\mathit{x}\mathbf{\ge }\mathbf{3}$

H $\mathit{x}\mathbf{<}\mathbf{-}\mathbf{2}$or $\mathit{x}\mathbf{\ge }\mathbf{3}$

J $\mathbf{-}\mathbf{2}\mathbf{<}\mathit{x}\mathbf{\le }\mathbf{3}$

Short answer

The compound inequality for the given graph is$\mathit{x}\mathbf{\le }\mathbf{-}\mathbf{2}$or $\mathit{x}\mathbf{\ge }\mathbf{3}$.Therefore, the option G is correct.

Step-by-step solution

Step 1. Observe the given graph.

The given graph is:

From the given graph, it can be noticed that on the left hand side of the number line, the graph is denoting numbers less than or equal to $-2$as at $-2$, there is a closed dot.

From the given graph, it can also be noticed that on the right hand side of the number line, the graph is denoting numbers greater than or equal to 3 as at 3, there is a closed dot.

The graph is showing the union of the numbers less than or equal to $-2$and numbers greater than or equal to 3.

Step 2. Write a compound inequality for the given graph.

The compound inequality for the given graph is the union of the numbers less than or equal to $-2$and numbers greater than or equal to 3.

A compound inequality containing ‘or’ is true if one or both inequalities are true.

That implies the solution of the compound inequality containing ‘or’ is the union of the solutions of the two simple statements.

Therefore, as the compound inequality for the given graph is the union of the numbers less than or equal to $-2$and numbers greater than or equal to 3.

Therefore, the compound inequality for the given graph is numbers less than or equal to $-2$or numbers greater than or equal to 3.

So, $x\le -2$or $x\ge 3$.

Therefore, the compound inequality for the given graph is $x\le -2$or $x\ge 3$.

Thus, the option G is correct.