Solve the compound inequality: 5x - 10 ≤ -55 or -35 ≤ 5x - 10
Understand the Problem
The question is asking us to solve a compound inequality and express the solution set. We need to solve each inequality separately and then combine the solutions using 'or'.
Answer
$(-\infty, -2) \cup (-1, \infty)$
Answer for screen readers
$(-\infty, -2) \cup (-1, \infty)$
Steps to Solve
- Solve the first inequality
Solve $2x + 7 < 3$ for $x$: $2x + 7 < 3$ $2x < 3 - 7$ $2x < -4$ $x < -2$
- Solve the second inequality
Solve $-3x - 5 < -2$ for $x$: $-3x - 5 < -2$ $-3x < -2 + 5$ $-3x < 3$ $x > -1$ (Remember to flip the inequality sign when dividing by a negative number)
- Express the solution set
The solution set is $x < -2$ or $x > -1$. In interval notation, this is $(-\infty, -2) \cup (-1, \infty)$.
$(-\infty, -2) \cup (-1, \infty)$
More Information
The solution set includes all numbers less than -2 and all numbers greater than -1. There is a gap between -2 and -1, which are not included in the solution.
Tips
A common mistake is forgetting to flip the inequality sign when dividing by a negative number, as in the second inequality. Another common mistake is incorrectly combining the two inequalities, especially when dealing with "or" conditions. Also, interval notation can be tricky so be aware of when to use parentheses versus brackets.
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