Problem 33 - Precalculus Student Solutions Manual 5th

Question

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

Step-by-Step Solution

Answer

(4, 6)

Step 1: Break the compound inequality

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

Step 2: Solve the first inequality

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

Step 3: Solve the second inequality

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

Step 4: Combine solutions

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

Step 5: Write the solution in interval notation

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

Key Concepts

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.