Problem 41 - Elementary Algebra

Question

For the following problems, what numbers can replace \(x\) so that the following statements are true? \(-1 \leq x \leq 5 \quad x\) an int eger

Step-by-Step Solution

Answer

Answer: The integer values of x within the given range are -1, 0, 1, 2, 3, 4, and 5.

Step 1: Identify the range for x

The given range for x is -1 to 5, inclusive. This means we need to find all integers between -1 and 5, including the endpoints.

Step 2: List all integers within the range

We will now list all integer values within our given range of \(-1 \leq x \leq 5\). -1, 0, 1, 2, 3, 4, 5

Step 3: Identify the values of x that make the statement true

Since our range includes all integers between -1 and 5, we've found all the possible values of x that would satisfy the statement. The values of x that can replace the variable and make the statement true are: -1, 0, 1, 2, 3, 4, and 5.

Key Concepts

Integer Solutions

When solving algebraic problems, particularly inequalities that specify that the solutions must be integers, our approach differs slightly from dealing with general real numbers. This is because integers are whole numbers that do not include fractions or decimals. In the context of the given exercise, where we seek to find integer solutions within the inequality eq -1 leq x leq 5code>, it is essential to note that we are not just looking for any number between -1 and 5, but specifically the whole numbers that fall within this range.

The process generally involves identifying the lower and upper bounds - which, in our case, are -1 and 5, respectively - and simply listing all integers that lie between these two values, inclusive. This straightforward method ensures that all possible integer solutions are considered without missing any potential candidates. In applications beyond textbook exercises, recognizing integer constraints is crucial for problems in number theory, computer science, and scenarios where fractional or continuous answers do not make sense in practical terms.

Range of Variable

In algebra, the 'range of variable' refers to the set of values that a variable can take on. It is pivotal for problem-solving because it provides the bounds within which solutions must lie. Inequalities often determine the range, as seen in the example exercise given. To solve for the range of a variable efficiently, first interpret the inequality symbols. For instance, leqcode> sign symbolizes 'less than or equal to', and it means that the variable value should not exceed the upper limit. Conversely, geqcode> means 'greater than or equal to' and sets the minimum limit for the variable's value.

Understanding the range is important as it impacts the strategy to be used for finding solutions. When dealing with linear inequalities, sketching a number line can help visually represent the range, making it clearer which values are included or excluded. The given range for our exercise was between -1 and 5, and by acknowledging the inclusive nature of the inequality, we could concisely note that the range of variable 'x' contains all the integer points within and including the endpoints of this segment.

Solving Inequalities

Solving inequalities is one of the fundamental skills in algebra. Unlike equations, which express equality, inequalities show a relation of less than, greater than, less than or equal to, or greater than or equal to between two expressions. To solve an inequality, you start by isolating the variable of interest on one side of the inequality symbol. This process can involve various operations such as adding, subtracting, multiplying, or dividing both sides of the inequality by the same nonzero number. However, an important aspect to remember is that if you multiply or divide by a negative number, the inequality sign has to be flipped.

Once the variable is isolated, you determine the range of values that satisfies the inequality. This part of the process may also include checking for integer solutions if the problem specifies this requirement or considering the entire spectrum of real numbers when no such constraints are present. In the given exercise, the problem already presents a defined inequality eq -1 leq x leq 5code>, so the solving part is straightforward - we identify which values of 'x' make the inequality true and match the criteria given, e.g., being an integer in this case.