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Chapter 1: Problem 33

Solve each inequality. Write each solution set in interval notation. $$-11>-3 x+1>-17$$

Short Answer

Expert verified
(4, 6)

Step by step solution

01

Break the compound inequality

Write the given compound inequality as two separate inequalities: \(-11 > -3x + 1\) and \(-3x + 1 > -17\).
02

Solve the first inequality

For the first inequality, \(-11 > -3x + 1\): \[ -11 - 1 > -3x \ -12 > -3x \ 4 < x (\text{Dividing both sides by } -3 \text{and reversing the inequality})\]
03

Solve the second inequality

For the second inequality, \(-3x + 1 > -17\): \[ -3x + 1 > -17 \ -3x > -18 (\text{Subtracting 1 from both sides}) \ x < 6 (\text{Dividing both sides by } -3 \text{and reversing the inequality})\]
04

Combine solutions

Combine the solutions from steps 2 and 3. The solution to the compound inequality is the overlap of both individual solutions: \[4 < x < 6\]
05

Write the solution in interval notation

The solution \(4 < x < 6\) in interval notation is \((4, 6)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

compound inequalities
Compound inequalities are mathematical expressions that combine two or more inequalities connected by the words 'and' or 'or.'. They help describe a range of possible solutions. For example, in the inequality e.g., $$-11 > -3x +1 > -17 $$we are saying $$-3x + 1 $$ must be greater than $$-17$$ AND less than $$ -11$$.
To solve it, we need to separate it into two individual inequalities:
First: $$ -11 > -3x + 1 $$ Second: $$ -3x + 1 > -17 $$.
interval notation
Interval notation is a way to describe the set of solutions to an inequality. It uses brackets and parentheses to show the range of numbers included in the solution set. For example, a solution $$4 < x < 6$$ in interval notation is written as (4, 6).
The round parentheses indicate that the endpoints, 4 and 6, are not included in the solution. If an endpoint was included (closed interval), we would use square brackets, like this: [4, 6].
Interval notation is a concise way to write solution sets, making it easier to understand the range of possible solutions at a glance.
reversing inequality
When solving inequalities, especially when dividing or multiplying by a negative number, the direction of the inequality must be reversed. For example:
If we have $$-3x > -18$$
and we divide both sides by $$-3$$ we reverse the inequality, resulting in
$$x < 6$$.
Reversing the inequality ensures that the relationship between the two sides remains accurate. It's crucial to remember this step to avoid mistakes and find the correct solution.

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