Problem 37 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Write each solution set in interval notation. $$-3 \leq \frac{x-4}{-5}<4$$

Step-by-Step Solution

Answer

The solution set is o\( (-16, 19] \).

Step 1: Break down the compound inequality

Split the compound inequality into two separate inequalities: o\( -3 \leq \frac{x-4}{-5} \) and o\( \frac{x-4}{-5} < 4 \)

Step 2: Solve the first inequality

Isolate the fraction by multiplying both sides by o\( -5 \). This reverses the inequality:o\( -3 \times -5 \geq (x-4) \rightarrow 15 \geq x-4 \). Finally, add 4 to both sides: o\( 15 + 4 \geq x \rightarrow 19 \geq x \)

Step 3: Solve the second inequality

Isolate the fraction by multiplying both sides by o\( -5 \). This reverses the inequality:o\( \frac{x-4}{-5} < 4 \times -5 \rightarrow x - 4 > -20 \). Finally, add 4 to both sides: o\( x - 4 + 4 > -20 + 4 \rightarrow x > -16 \)

Step 4: Combine the results

Combine the solutions from both inequalities: o\( -16 < x \leq 19 \). The solution set in interval notation is: o\( (-16, 19] \)

Key Concepts

Compound Inequalities

Compound inequalities involve two separate inequalities that are connected by either an 'and' or an 'or'. In the given problem, we have a compound inequality:

• \(-3 \leq \frac{x-4}{-5}\< 4\)

This type is connected by an 'and' because we need to solve for values of \(x\) that satisfy both parts of the inequality simultaneously.
To tackle this, we can break it down into two separate inequalities:

• \(-3 \leq \frac{x-4}{-5}\)
• \(\frac{x-4}{-5}\< 4\)

By solving each inequality individually, we find the range of values for \(x\) that make both conditions true. Finally, we can combine the solutions to find the value range for \(x\).
Understanding compound inequalities helps to simplify complex problems and is critical in various mathematical applications.

Interval Notation

Once you solve a compound inequality, expressing the solution in interval notation makes it more concise and easier to understand.
Interval notation uses brackets and parentheses to describe the range of solutions in terms of the lower and upper bounds.

• \((a, b)\): Values between \(a\) and \(b\), not including \(a\) and \(b\).
• \([a, b]\): Values between \(a\) and \(b\), including both \(a\) and \(b\).
• \((a, b]\): Values between \(a\) and \(b\), including \(b\) but not \(a\).

In our problem, the solution set \(-16 < x \leq 19\) is converted to interval notation as \((-16, 19]\).
This method is efficient for representing multiple ranges and understanding solution sets more intuitively.

Reverse Inequalities

When solving inequalities, multiplying or dividing by a negative number reverses the inequality sign. This is an essential rule that ensures that the inequality remains accurate.
For example, in our problem, we must reverse the inequality signs when we multiply by -5:

• \(-3 \leq \frac{x-4}{-5}\)
• Multiplying both sides by -5 reverses the inequality to: \(15 \geq x-4\)
• So, \(x\leq 19\)

Likewise, for \(\frac{x-4}{-5} \lt 4\), multiplying by -5 reverses the sign:

• \(\frac{x-4}{-5} \lt 4\)
• Multiplying both sides by -5 makes it: \(x-4\gt -20\)
• So, \(x \gt -16\)

Keeping track of these changes is crucial to avoid errors in solving inequalities and finding the correct solution sets.