Solve the compound inequality: 3x - 5 < -23 or -14 ≤ 3x - 5

Understand the Problem

The question is asking us to solve a compound inequality that involves two inequalities connected by "or". We need to solve each inequality separately and then combine the solutions.

Answer

$(-\infty, -2) \cup (3, \infty)$

Steps to Solve

Solve the first inequality

We have $2x + 1 < -3$. Subtract 1 from both sides:

$2x < -3 - 1$

$2x < -4$

Divide both sides by 2:

$x < -2$

Solve the second inequality

We have $3x - 5 > 4$. Add 5 to both sides:

$3x > 4 + 5$

$3x > 9$

Divide both sides by 3:

$x > 3$

Combine the solutions

The solution to the compound inequality is $x < -2$ or $x > 3$. In interval notation, this is $(-\infty, -2) \cup (3, \infty)$.

$(-\infty, -2) \cup (3, \infty)$

More Information

The solution represents all real numbers that are either less than -2 or greater than 3. There is a gap between -2 and 3, meaning that numbers in that range are not part of the solution.

Tips

A common mistake is to incorrectly perform the algebraic manipulations when solving each inequality, such as forgetting to flip the inequality sign when multiplying or dividing by a negative number (though this wasn't relevant in this specific problem).. Another common mistake is to misunderstand the meaning of "or" and "and" in compound inequalities and incorrectly combine the solution sets. In this case, "or" means that the solution includes all values that satisfy either inequality.

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