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Chapter 9: Problem 20

Solve each inequality, and graph the solution on the number line. $$ 3 x-9<-4.5 x+6 $$

Short Answer

Expert verified
x < 2

Step by step solution

01

- Combine Like Terms

First, move the terms involving x to one side of the inequality. Add \( 4.5x \) to both sides: \[ 3x + 4.5x - 9 < 6 \] This simplifies to: \[ 7.5x - 9 < 6 \]
02

- Isolate the Variable Term

Next, isolate the term with x by adding 9 to both sides: \[ 7.5x - 9 + 9 < 6 + 9 \] Which simplifies to: \[ 7.5x < 15 \]
03

- Solve for x

Now, divide both sides of the inequality by 7.5 to solve for x: \[ x < \frac{15}{7.5} \] Simplifying gives: \[ x < 2 \]
04

- Graph the Solution

To graph the solution on a number line, draw a number line, and place an open circle at 2 (since x is less than 2, but not equal to 2). Then, shade the region to the left of 2 to indicate that x can be any number less than 2.

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Key Concepts

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

algebraic inequalities
Algebraic inequalities are statements that show the relationship between two expressions using inequality symbols such as <, >, 鈮�, and 鈮�. In general, when solving inequalities, the goal is to isolate the variable on one side of the inequality.

For example, consider the inequality from our exercise: 3x - 9 < -4.5x + 6. The steps to solve this involve combining like terms and isolating the variable, followed by interpreting and graphing the solution.

Remember that when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed. This is a crucial point to keep in mind to avoid errors in solutions.
combining like terms
Combining like terms is an essential step in solving inequalities. Like terms are terms that contain the same variable raised to the same power. To simplify an expression, these terms are added or subtracted.

Let's look at our inequality: 3x - 9 < -4.5x + 6. Our first step is to get all the x terms on one side and the constant terms on the other side. By adding 4.5x to both sides, we combine the x terms: 3x + 4.5x - 9 < 6. This simplifies to: 7.5x - 9 < 6. Now, the x terms are combined into a single term, making it easier to proceed to the next steps.
graphing solutions
Graphing the solution of an inequality helps visualize the range of possible values for the variable. For our exercise, after solving the inequality, we find that x < 2.

To graph this, we use a number line:

1. Draw a horizontal line and mark a point for the number 2.
2. Place an open circle on the number 2 to show that 2 is not included in the solution (since x is less than 2, not less than or equal to 2).
3. Shade the region to the left of the open circle to indicate all numbers less than 2. This shaded area represents all the possible values of x that satisfy the inequality.

Graphing is a powerful tool as it provides a clear visual representation of the solution set.

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Most popular questions from this chapter

All the whole-number pairs with a sum of 26 are put into a hat, and one is drawn at random. a. List all the possible whole-number pairs with a sum of 26. b. What is the size of the sample space in this situation? c. What is the probability that at least one of the numbers in the pair selected is greater than or equal to 15? Explain how you found your answer. d. What is the probability that both numbers in the pair selected are less than 15? Explain.

Simplify each expression. $$ \frac{7}{k-2}-\frac{5}{2(k-2)} $$

Suppose two people play Rolling Differences with two 8 -sided dice numbered 1 to \(8 .\) The players follow these rules: \(\cdot\) Player 1 scores 1 point if the difference is \(0,1,\) or \(2 .\) \(\cdot\) Player 2 scores 1 point if the difference is \(3,4,5,6,\) or 7 Which player has the advantage? Explain your answer.

Suppose the \(2-\) of \(-6\) lottery game was modified so that after the first number was selected, that number was placed back into the group before the next number was selected. In this way, a number could be repeated, meaning pairs such as \(2-2\) and \(3-3\) would be possible. a. Would your chances of winning be better or worse for this modified game? Explain. b. How many possible pairs are there for this modified game, assuming that order does matter? Explain. c. List all of the possible pairs from part b. d. Since order really doesn't matter in this game, how many different pairs are there? (Remember, if order doesn't matter, \(1-2\) is the same as \(2-1 .\) ) e. Are all of the pairs considered in part d equally likely? Explain. f. If you choose one number pair for this modified game, what is the probability you will win. (Hint: There are two cases to consider.)

A programmer wrote some software that composes pieces of music by randomly combining musical segments. For each piece, the program randomly chooses 4 different segments from a group of 20 possible segments and combines them in a random order. How many different musical pieces can be created in this way? (Hint: How many choices are there for the first segment? For each of those, how many choices are there for the second segment?)
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