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Chapter 1: Problem 95

Solve the inequality. Write the solution set in interval notation. $$6<-\frac{1}{2} p+4$$

Short Answer

Expert verified
(-鈭�, -4)

Step by step solution

01

- Isolate the variable term

Firstly, subtract 4 from both sides of the inequality to isolate the term involving the variable on one side. 6 - 4 < -\( \frac{1}{2} \) p + 4 - 4 This simplifies to:2 < -\( \frac{1}{2} \) p
02

- Eliminate the fraction

Multiply both sides of the inequality by -2 to eliminate the fraction. Remember to reverse the inequality sign when multiplying or dividing by a negative number:2 \times (-2) > -\( \frac{1}{2} \) p \times (-2) This simplifies to: -4 > p
03

- Write the solution in interval notation

Rewrite the inequality as p < -4 and express this solution in interval notation: (-鈭�, -4)

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Key Concepts

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

Isolate the Variable
When solving inequalities, we aim to get the variable by itself on one side of the inequality sign. This process is called isolating the variable.

In the given exercise, our inequality is: 6 < -\( \frac{1}{2} \) p + 4.
To isolate the variable term (which here is -\( \frac{1}{2} \) p), we perform the following steps:
  • Subtract 4 from both sides to cancel out the +4 on the right-hand side.

This step distributes as follows: $$ 6 - 4 < -\( \frac{1}{2} \) p + 4 - 4$$ Simplifying this, we get: $$ 2 < -\( \frac{1}{2} \) p $$
Now, the inequality is simplified and the variable term has been isolated.
Inequality Sign Reversal
A key detail when solving inequalities is knowing when to reverse the inequality sign. This reversal occurs when you multiply or divide both sides of the inequality by a negative number.

In our example, we need to get rid of the fraction, -\( \frac{1}{2} \). We do so by multiplying both sides by -2. Remember, since we are multiplying by a negative number, we must reverse the inequality sign:

$$ 2 (-2) > -\( \frac{1}{2} \) p (-2) $$
Simplifying, we find: $$ -4 > p $$
We reversed the inequality sign from '<' to '>' because of the negative multiplication.
Interval Notation
Once you have the solution to the inequality, it's important to express it properly. One way to do this is using interval notation.

In our exercise, the solution, after simplification, is: $$ p < -4 $$

Interval notation is a way to describe a set of numbers along the number line. For 'p < -4', this means all values of p that are less than -4.

In interval notation, this is written as:
$$ (-鈭�, -4) $$
The round brackets indicate that -鈭� and -4 are not included in the set. This notation helps to easily understand and communicate the range of values that satisfy the inequality.

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Most popular questions from this chapter

Write an absolute value inequality that represents the statement. \(7-y<-1\) or \(7-y>1\)

Explain why the inequality \(|x|>-5\) is true for all real numbers \(x\).

Suppose that over the course of 3 months, a court system selects 1000 jurors from a large jury pool having an equal number of males and females. If males and females are selected at random, there should be approximately 500 males and 500 females selected. However, these values may vary slightly. If 1000 people are selected at random from the large jury pool, the inequality \(\left|\frac{x-500}{\sqrt{250}}\right|<1.96\) gives the "reasonable" range for the number of women selected, \(x\). a. Solve the inequality and interpret the answer in the context of this problem. (Hint: Round so that the endpoints of the interval are whole numbers, but still within the interval defined by the inequality.) b. If the group of 1000 jurors has 560 women, does it appear that there may be some bias toward women jurors?

Explain why \(x^{2}=4\) is equivalent to the equation \(|x|=2\).

Solve the inequality, and write the solution set in interval notation. \(\left|\frac{m-4}{2}\right|<14\)
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