Is the inequality 22x + 9 < 4x + 9 sometimes, always, or never true?

To determine whether the inequality 22x + 9 < 4x + 9 is sometimes, always, or never true, we can start by simplifying the inequality.

First, we can subtract 4x from both sides:

22x – 4x + 9 < 9

This simplifies to:

18x + 9 < 9

Next, let’s subtract 9 from both sides:

18x < 0

Now, divide both sides by 18:

x < 0

This result tells us that the inequality 22x + 9 < 4x + 9 is true for all values of x that are less than 0.

Therefore, the inequality is sometimes true—specifically, it holds true when x is negative. If x is greater than or equal to 0, the inequality does not hold. Hence, we conclude that the inequality is not universally valid, but rather conditional based on the value of x.