Problem 49 - Elementary Algebra

Question

For the following problems, what integers can replace \(x\) so that the statements are true? $$ -3

Step-by-Step Solution

Answer

Answer: The integer values of x that satisfy the inequality are -2, -1, 0, 1, and 2.

Step 1: Analyze the inequality

The inequality we are given is \(-3

Step 2: Identify the integers that fit the inequality

Recall that integers are whole numbers, including negative whole numbers, zero, and positive whole numbers. We need to identify the integers that are greater than \(-3\) and less than or equal to \(2\). Looking at a number line, we find the following integers fit this description: -2, -1, 0, 1, and 2 Make sure not to include -3, because the inequality states \(x\) must be greater than -3. So, the integers that can replace \(x\) so that the inequality is true are \(\boxed{-2, -1, 0, 1, 2}\).

Key Concepts

Integer Solutions

When solving inequalities that ask for integer solutions, it's important to understand that integers constitute the set of whole numbers, which includes positive numbers, negative numbers, and zero. However, numbers with fractional or decimal parts are not considered integers.

To find integer solutions for an inequality like \( -3 < x \leq 2 \), we seek out all the whole numbers that fall between the specified range. Remember, the inequality sign points to the number which is smaller, and if it has a line beneath it (\leq), it includes that number as well. In this case, since the inequality does not include -3 (indicated by the < sign), but does include the 2 (indicated by \leq), the integer solutions are -2, -1, 0, 1, and 2.

It is crucial to pay attention to these signs while identifying the correct set of integers that meet the conditions of the inequality.

Inequality Analysis

Inequality analysis is a systematic approach to solving inequalities. This particular exercise involved analyzing a simple linear inequality \( -3 < x \leq 2 \), which is read as 'x is greater than -3 and less than or equal to 2.' It's crucial to understand each part of the inequality:

• The < symbol indicates that -3 is not part of our solution.
• The \leq symbol means 2 is included.

This distinction dictates the approach one needs to take when listing potential solutions. Through careful analysis, we ensure that our answers are both complete and within the bounds set by the exercise. Inequalities require us to consider a range of numbers rather than a precise value, as is the case with equations. As such, being methodical in interpreting each part of an inequality is essential.

Using A Number Line

A number line is an excellent tool for visualizing inequalities and identifying integer solutions. It represents numbers as points on a straight line where each point corresponds to a number.

In our exercise, plotting a number line with a clear mark for -3 and 2 would be the starting point. Then, identify -3 with an open circle to show that -3 is not included, and 2 with a closed or filled-in circle to signal that it is included in the solutions. Draw a line or arrow between these two points to represent all numbers that fall within this range.

This visual aid makes it significantly easier to understand which integers are part of the solution set, especially when dealing with negative numbers or larger ranges. It's a simple yet effective way to ensure accuracy when tackling inequalities.

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Problem 49

Other exercises in this chapter

Problem 48

Is every integer a rational number?

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Problem 48

For the following problems, rewrite each phrase using algebraic notation. \(x\) divided by \((7\) times \(b)\)

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Problem 49

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbe

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Problem 49

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$

View solution

Key Concepts

Integer Solutions

When solving inequalities that ask for integer solutions, it's important to understand that integers constitute the set of whole numbers, which includes positive numbers, negative numbers, and zero. However, numbers with fractional or decimal parts are not considered integers.

To find integer solutions for an inequality like \( -3 < x \leq 2 \), we seek out all the whole numbers that fall between the specified range. Remember, the inequality sign points to the number which is smaller, and if it has a line beneath it (\leq), it includes that number as well. In this case, since the inequality does not include -3 (indicated by the < sign), but does include the 2 (indicated by \leq), the integer solutions are -2, -1, 0, 1, and 2.

It is crucial to pay attention to these signs while identifying the correct set of integers that meet the conditions of the inequality.

Inequality Analysis

Inequality analysis is a systematic approach to solving inequalities. This particular exercise involved analyzing a simple linear inequality \( -3 < x \leq 2 \), which is read as 'x is greater than -3 and less than or equal to 2.' It's crucial to understand each part of the inequality:

• The < symbol indicates that -3 is not part of our solution.
• The \leq symbol means 2 is included.

This distinction dictates the approach one needs to take when listing potential solutions. Through careful analysis, we ensure that our answers are both complete and within the bounds set by the exercise. Inequalities require us to consider a range of numbers rather than a precise value, as is the case with equations. As such, being methodical in interpreting each part of an inequality is essential.

Using A Number Line

A number line is an excellent tool for visualizing inequalities and identifying integer solutions. It represents numbers as points on a straight line where each point corresponds to a number.

In our exercise, plotting a number line with a clear mark for -3 and 2 would be the starting point. Then, identify -3 with an open circle to show that -3 is not included, and 2 with a closed or filled-in circle to signal that it is included in the solutions. Draw a line or arrow between these two points to represent all numbers that fall within this range.

This visual aid makes it significantly easier to understand which integers are part of the solution set, especially when dealing with negative numbers or larger ranges. It's a simple yet effective way to ensure accuracy when tackling inequalities.