Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the values of \(x\) that satisfy the inequalities. $$ 0 \leq x+1 \leq 4 $$

Step-by-Step Solution

Answer

The values of \(x\) that satisfy the inequalities are \(-1 \leq x \leq 3\).

Step 1: Rewrite the inequality

Write the given chained inequality as two separate inequalities: \(0 \leq x+1\) and \(x+1 \leq 4\).

Step 2: Solve the first inequality

Solve the inequality \(0 \leq x+1\) for x. To do this, subtract 1 from each side of the inequality: \(0-1 \leq x+1-1\), which gives \(-1 \leq x\).

Step 3: Solve the second inequality

Now, solve the inequality \(x+1 \leq 4\) for x. To do this, subtract 1 from each side of the inequality: \(x+1-1 \leq 4-1\), which gives \(x \leq 3\).

Step 4: Combine the inequalities

Combine the inequalities from Steps 2 and 3: \(-1 \leq x \leq 3\). So the values of x that satisfy both inequalities are the values between -1 (inclusive) and 3 (inclusive).

Key Concepts

Linear Inequalities

Understanding linear inequalities is crucial for solving problems that require finding a range of solutions, rather than a single answer. A linear inequality looks much like a linear equation, but instead of an equal sign (\(=\)), it includes an inequality symbol such as 'greater than' (\(>\)), 'less than' (\(<\)), 'greater than or equal to' (\(≥\)), or 'less than or equal to' (\(≤\)). These symbols indicate a range of possible solutions.

For instance, the inequality from the exercise, \(0 ≤ x+1 ≤ 4\), represents a linear inequality where \(x\) can take on any value within an interval. The process of solving this kind of inequality involves isolating the variable \(x\) on one side of the inequality to determine this range. Just like with equations, you can add, subtract, multiply, or divide both sides of an inequality by the same nonzero number without changing the inequality's direction, unless you multiply or divide by a negative number, which reverses the inequality sign.

Solving linear inequalities is very much like solving linear equations, but you end up with a range of \(x\) values that fulfill the inequality rather than a single value.

Inequality Notation

Inequality notation is the shorthand method of writing inequalities that efficiently communicate the information about the relationship between values. With the exercise provided, we have a compound inequality: \(0 ≤ x+1 ≤ 4\). This is known as a 'chained' inequality since it chains together two inequalities. Chained inequalities are a compact way of expressing that a variable is within a specified range.

The 'chained' form combines the conditions that \(x+1\) must be greater than or equal to 0 and less than or equal to 4. The notation is merged, which shows us that \(x\) itself has to lie between the values of -1 and 3, as found by solving the two separate inequalities. Using the square brackets or parentheses can also denote whether the endpoints are included (\(≤, ≥\) ) or not (\(>, <\) ). In our case, square brackets would be used to signify that \(x\) can be exactly -1 or 3, hence: \[ -1, 3 \].

Mathematical Reasoning

Mathematical reasoning involves logically deducing the steps to solve a problem and comprehending how individual steps contribute to the solution. Throughout the process of solving inequalities, mathematical reasoning is used to decide which operations to apply and in what order. In our exercise example, it's about recognizing that to isolate \(x\), one must undo the addition of 1 by subtracting it from all parts of the chained inequality.

The reasoning further comes into play when combining the results of the separate inequalities. By recognizing that \(x\) must satisfy both \( -1 ≤ x \) and \( x ≤ 3 \) concurrently, we use logical deduction to conclude that the solution set is the intersection of both ranges, which gives us the final answer that \(x\) can be any real number between -1 and 3 inclusive. Mathematical reasoning is, therefore, the key to understanding not just the