# How do you solve 4x + 1 ≤ 17 and 5x - 1 > 19?

Jul 7, 2015

This compound inequality has no solutions.

#### Explanation:

To solve a compound inequality consisting of two inequalities joined by the word "and" we need to find all values of $x$ that make both inequalities true. That is: he same $x$ has to make both true.

Solving $4 x + 1 \le 17$, we get:
$4 x \le 16$, so $x \le 4$

Solving $5 x - 1 > 19$, we get $x > 4$

To solve: "$4 x + 1 \le 17 \text{ and } 5 x - 1 > 19$", we need all $x$'s that solve

"$x \le 4 \text{ and } x > 4$"

That is, we need an $x$ that is, at the same time, both less than or equal to $4$ and also strictly greater than $4$.

There is no such $x$, so the compound inequality has no solution.

(If you need to write the solution set, use the notation for the empty set: "$\emptyset$")