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Chapter 1: Problem 82

Answer true or false given that \(a>0, b<0, c>0,\) and \(d<0\). $$a b

Short Answer

Expert verified
True

Step by step solution

01

Understand the signs of the variables

Given that: - \(a > 0\) which means \(a\) is positive.- \(b < 0\) which means \(b\) is negative.- \(c > 0\) which means \(c\) is positive.- \(d < 0\) which means \(d\) is negative.
02

Determine the sign of the product \(ab\)

Since \(a\) is positive and \(b\) is negative, the product \(ab\) will be negative. A positive value multiplied by a negative value results in a negative value.
03

Evaluate \(c\)

Given that \(c > 0\), this implies that \(c\) is positive.
04

Compare the product \(ab\) with \(c\)

Since \(ab\) is negative (as determined in Step 2) and \(c\) is positive (as determined in Step 3), a negative value (\(ab\)) is always less than a positive value (\(c\)).
05

Conclusion

Based on the comparison in Step 4, the statement \(ab < c\) is true.

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Key Concepts

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

Sign of Products
When working with algebra, understanding the sign of products is crucial. The sign of a product depends on the signs of the numbers being multiplied. Here are some key points to remember:
  • If you multiply two positive numbers, the product is positive.
  • If you multiply two negative numbers, the product is positive.
  • If you multiply a positive number and a negative number, the product is negative.
Let's consider the example from the exercise: we know that \(a > 0\) (positive) and \(b < 0\) (negative). When we multiply these two, the product \(ab\) will be negative. This is confirmed in the step-by-step solution provided in the exercise. Remembering these simple rules will help you determine the sign of any product, which is an essential skill in solving inequalities.
Inequality Comparison
Inequalities allow us to compare two values to see which one is greater, smaller, or if they are equal. The symbol \(<\) means 'less than', and the symbol \(>\) means 'greater than'.

In the given exercise, we are to determine whether \(ab < c\). We've already established that \(ab\) (a positive number times a negative number) is negative. Knowing that \(c > 0\) means \(c\) is positive, it becomes easier to compare.

In general:
  • A negative number is always less than a positive number.
  • This means that if you have a negative product and a positive number, the negative product will always be smaller.
Using this information, it is clear that \(ab < c\) is true because every negative number is less than any positive number. Always check the signs before comparing values in an inequality.
Positive and Negative Numbers
Understanding positive and negative numbers is fundamental to solving any algebra problem. Here's a brief refresher:
  • Positive numbers are greater than zero (e.g. 1, 2, 3).
  • Negative numbers are less than zero (e.g. -1, -2, -3).
When adding or subtracting these numbers, remember:
  • Adding two positive numbers gives a positive result.
  • Adding two negative numbers gives a negative result.
  • Subtracting a positive number from another positive number depends on their absolute values.
  • Subtracting a negative number is the same as adding its positive counterpart.
These rules extend to multiplication and divisions as well. In the exercise, understanding that \(a > 0\) and \(b < 0\) is key to concluding that \(ab\) will be negative, and consequently, less than \(c\), which is positive. Paying close attention to the signs of numbers ensures accurate inequality comparisons.

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Most popular questions from this chapter

Seismographs can record two types of wave energy (P waves and S waves) that travel through the Earth after an earthquake. Traveling through granite, \(\mathrm{P}\) waves travel approximately \(5 \mathrm{~km} / \mathrm{sec}\) and \(\mathrm{S}\) waves travel approximately \(3 \mathrm{~km} / \mathrm{sec} .\) If a geologist working at a seismic station measures a time difference of 40 sec between an earthquake's P waves and S waves, how far from the epicenter of the earthquake is the station?

A 10 -yr-old competes in gymnastics. For several competitions she received the following "All-Around" scores: \(36,36.9,37.1,\) and \(37.4 .\) Her coach recommends that gymnasts whose "All-Around" scores average at least 37 move up to the next level. What "All-Around" scores in the next competition would result in the child being eligible to move up?

a. Write an absolute value equation or inequality to represent each statement. b. Solve the equation or inequality. Write the solution set to the inequalities in interval notation. The distance between a number \(x\) and 3 on the number line is 8 .

a. Write an equation representing the fact that the sum of the squares of two consecutive integers is 181 . b. Solve the equation from part (a) to find the two integers.

Use the model \(s=-\frac{1}{2} g t^{2}+v_{0} t+s_{0}\). In a classic Seinfeld episode, Jerry tosses a loaf of bread (a marble rye) straight upward to his friend George who is leaning out of a third-story window. a. If the loaf of bread leaves Jerry's hand at a height of \(1 \mathrm{~m}\) with an initial velocity of \(18 \mathrm{~m} / \mathrm{sec},\) write an equation for the vertical position of the bread \(s\) (in meters) \(t\) seconds after release. b. How long will it take the bread to reach George if he catches the bread on the way up at a height of \(16 \mathrm{~m} ?\) Round to the nearest tenth of a second.
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