Problem 17 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the values of \(x\) that satisfy the inequalities. $$ x+1>4 \text { or } x+2<-1 $$

Step-by-Step Solution

Answer

The values of x that satisfy the inequalities are \(x > 3\) or \(x < -3\).

Step 1: Solve the first inequality: x + 1 > 4

To solve x + 1 > 4, we need to isolate x. Subtract 1 from both sides of the inequality like this: \(x + 1 - 1 > 4 - 1\) This simplifies to: \(x > 3\) So, the range of values of x that satisfy this inequality is \(x > 3\).

Step 2: Solve the second inequality: x + 2 < -1

To solve x + 2 < -1, we need to isolate x. Subtract 2 from both sides of the inequality like this: \(x + 2 - 2 < -1 - 2\) This simplifies to: \(x < -3\) So, the range of values of x that satisfy this inequality is \(x < -3\).

Step 3: Combine the solutions

We have two ranges of values for x that satisfy the given inequalities: \(x > 3\) and \(x < -3\). Since we are looking for values of x that satisfy either one of the inequalities or both, we can combine the two solutions like this: \(x > 3 \text{ or } x < -3\) This is the final answer, which represents the values of x that satisfy one or both of the given inequalities.

Key Concepts

Solving Linear Inequalities

Solving linear inequalities is similar to solving linear equations, but there is a key difference — inequalities do not just identify a single solution but a range of possible values. The goal is to isolate the variable, in this case, \(x\), on one side of the inequality sign.
To achieve this, we perform similar arithmetic operations as with equations: addition, subtraction, multiplication, or division. For instance, if we have an inequality like \(x + 1 > 4\), we subtract 1 from both sides to isolate \(x\), resulting in \(x > 3\).
It is crucial to remember when multiplying or dividing by a negative number, the inequality sign flips. So, if our step involves dividing by a negative number, we must reverse the direction of the inequality sign to maintain a true statement.

Mathematics Problem Solving

Problem solving in mathematics involves identifying the problem, planning a strategy for how to solve it, executing that plan, and finally, checking the solution for accuracy. In our specific problem, the task is to find all values of \(x\) that satisfy either of the two inequalities given: \(x+1>4\) or \(x+2<-1\).

• Understand the Problem: First, clearly identify each inequality. Here, we have two separate inequalities.
• Devise a Plan: Decide to solve each inequality independently. Then, combine their solutions since the logical connector "or" indicates that \(x\) can satisfy either inequality.
• Carry Out the Plan: Solve \(x + 1 > 4\) by subtracting 1 from both sides to get \(x > 3\), and \(x + 2 < -1\) by subtracting 2 to arrive at \(x < -3\).
• Check the Solution: Verify that the derived ranges for \(x\) indeed make the original inequalities true.

Logical Reasoning in Mathematics

Logical reasoning in mathematics is the ability to use systematic steps to solve problems and to justify why each step is taken. This involves critical thinking and knowing when to apply certain mathematical rules.

• Understanding "Or" in Inequalities: The word "or" in logical terms means that any condition provided can be true. In our exercise, the values of \(x\) that satisfy either \(x > 3\) or \(x < -3\) are the solution, making the entire statement true.
• Combining Solutions: When combining the solutions of each inequality, use logical reasoning to determine if both conditions can be true simultaneously. Here, they can't, because \(x\) cannot simultaneously be greater than 3 and less than -3.
• Checking the Entire Solution Set: Verify that the combined solution covers all the possible true scenarios based on the original inequalities.

Logical reasoning helps in ensuring our approach to solving inequalities is solid and that we've thoroughly examined all potential solutions.