Find the correct description of the graph of the compound inequality x − 3 < −9 or x + 5 ≥ 10.
Inequalities are very important concepts that are required in calculating the domain or the range of a variable in any algebraic, trigonometric, or differential equation.
Answer: The region represented by the given inequalities on the graph does not converge anywhere; hence, the given system of inequalities does not have any solution.
Let's understand in detail.
We are given the compound inequalities:
⇒ x − 3 < −9 (equation 1)
⇒ x + 5 ≥ 10 (equation 2)
We can rearrange equation 1 as:
⇒ x < 3 - 9
⇒ x < -6
We can rearrange equation 2 as:
⇒ x + 5 ≥ 10
⇒ x ≥ 10 - 5
⇒ x ≥ 5
Hence, we have the solutions as x < -6 and x ≥ 5.
Now, we can represent the above inequalities on the graph as shown below.
Here, x < -6 is represented by the red region on the left and x ≥ 5 is represented by the blue region on the right.
Now, from the graph above, we can see that the system of inequalities does not converge anywhere on the graph.
Hence, the region represented by the given inequalities on the graph does not converge anywhere; hence, the given system of inequalities does not have any solution.