Problem 48 - Elementary Algebra

Question

For the following problems, what integers can replace \(x\) so that the statements are true? $$ -3 \leq x<1 $$

Step-by-Step Solution

Answer

Answer: The possible integer values of x are -3, -2, -1, and 0.

Step 1: Analyze the inequality

The inequality consists of two parts: 1. \(-3 \leq x\): This part tells us that the integer \(x\) should be greater than or equal to -3. 2. \(x < 1\): This part tells us that the integer \(x\) should be less than 1. Now let's find the integers that satisfy both these conditions.

Step 2: Identify the integers that satisfy the inequality

To satisfy both conditions, \(x\) must be an integer that is greater than or equal to -3 and less than 1. The range of integers that meet these criteria are: $$-3, -2, -1, 0$$

Step 3: Provide the answer

The integers that can replace \(x\) so that the inequality is true are: $$-3, -2, -1, 0$$

Key Concepts

Algebraic Inequalities

Algebraic inequalities are mathematical expressions that show the relationship between two values, indicating that one is larger or smaller than the other. They are similar to equations but instead of an '=' sign, they use symbols like '\textless', '\textless=', '\textgreater', and '\textgreater='. For example, the inequality \( -3 \leq x \) states that \( x \) is any number greater than or equal to -3.

When dealing with inequalities, it's important to understand the symbols: '\textless=' means 'less than or equal to', while '\textgreater=' means 'greater than or equal to'. In our original exercise, \( -3 \leq x < 1 \) represents a compound inequality where \( x \) can be any integer between -3 and 1, including -3 but not 1. Understanding the meaning behind these symbols and how they connect values is crucial when resolving such mathematical problems.

Integer Number Line

The integer number line is a visual representation of integers in a fixed sequence, where each point corresponds to an integer. The line extends indefinitely in both the positive and negative directions. It's a helpful tool for visualizing the range of integer solutions for inequalities.

With our exercise, if you were to place -3 and 1 on the number line and highlight every integer between them, you'd mark -3, -2, -1, and 0. It's a straightforward way to see that these are all the integers that satisfy the inequality \( -3 \leq x < 1 \). Therefore, when trying to determine which integers can replace \( x \) in an inequality, envisioning them on the integer number line can be an effective method.

Solving Inequalities

Solving inequalities involves finding all possible values that satisfy the given inequality. The process is similar to solving equations, but the solutions are often ranges of numbers rather than a single number. Let's revisit our initial problem-step to better understand this process.

Firstly, translate the inequality into an easily understood statement. If \( -3 \leq x < 1 \), this means we're looking for all the integers where \( x \) is at least -3, but less than 1. The next step is to identify the integers that fall within this range, which we've established are -3, -2, -1, and 0. It's important to be careful when working with inequalities because the solution is a set of numbers, as opposed to the single solution often found in equations. Properly identifying and representing these solutions is key to mastering the art of solving inequalities.