Problem 47 - Elementary Algebra

Question

For the following problems, what integers can replace \(x\) so that the statements are true? $$ 4 \leq x \leq 7 $$

Step-by-Step Solution

Answer

Answer: The integer values of x that satisfy the inequality are 4, 5, 6, and 7.

Step 1: Analyze the given inequality

First, let's take a look at the inequality provided: $$ 4 \leq x \leq 7 $$ This inequality is telling us that x must be greater than or equal to 4 and at the same time less than or equal to 7.

Step 2: Identify the range of possible values for x

Given the inequality from Step 1, we can identify the range of possible values for x: $$ 4 \leq x \leq 7 $$ This means that x can take on any value between 4 and 7, inclusive.

Step 3: List out the integer values within the range

Now that we know the range of possible values for x, let's list out the integer values within that range: - When x = 4, the inequality is true (4 is between 4 and 7). - When x = 5, the inequality is true (5 is between 4 and 7). - When x = 6, the inequality is true (6 is between 4 and 7). - When x = 7, the inequality is true (7 is between 4 and 7). So, the integer values that make the inequality true are x = 4, x = 5, x = 6, and x = 7.

Key Concepts

Understanding Integer Solutions

In mathematics, integer solutions represent the solutions to an equation or inequality that are whole numbers. A whole number can be either positive or negative, but it does not include fractions or decimals. In the context of inequalities, integer solutions are those specific whole numbers that satisfy a given inequality. For example, in the inequality \( 4 \leq x \leq 7 \), values like 4, 5, 6, and 7 are the integer solutions because they are whole numbers that fit within the specified constraint. It is crucial to identify these values accurately so that any mathematical statements involving inequalities are satisfied.While real numbers could include all values between 4 and 7, listing integer solutions helps streamline any calculations or real-world applications requiring whole numbers. Therefore, when faced with an inequality, it's important to carefully identify if you are looking for all solutions or just integer ones. This distinction will guide you towards correct answers.

Exploring the Range of Values

In any mathematical inequality, the range of values refers to the span of numbers that can satisfy the inequality condition. The provided inequality \( 4 \leq x \leq 7 \) defines a clear range. Ranges are crucial because they clearly outline where solutions lie on a number line.For this inequality, the range of values is between 4 and 7. That means any number from 4 up to 7 can be a possible solution. Ranges can be defined for various forms of inequalities and are often expressed in interval notation, such as \([4, 7]\). Here, the brackets indicate that the endpoints, 4 and 7, are included in the range, which is a direct result of the inequality being inclusive.Visualizing the range on a number line can greatly aid understanding. It helps see where each valid solution rests in relation to the boundary numbers. Recognizing and determining these ranges are essential when solving inequalities.

Introduction to Inclusive Inequalities

Inclusive inequalities are those that include the boundary or endpoint numbers as part of the solution set. These are denoted by the symbols \( \leq \) or \( \geq \), which mean "less than or equal to" and "greater than or equal to," respectively. In other words, these inequalities do not exclude the numbers at their endpoints.In the example \( 4 \leq x \leq 7 \), the inclusion of 4 and 7 within the inequality symbols indicates that both 4 and 7 are part of the solution set. This is why they are called inclusive; every value from 4 to 7, including both 4 and 7 themselves, satisfies the inequality.Understanding inclusive inequalities is important because they define a closed range on a number line. Knowing the difference between inclusive and exclusive inequalities helps in correctly solving and graphing inequalities. With inclusive inequalities, you ensure that none of the potential solutions are inadvertently excluded, guaranteeing accuracy and completeness in solving.