Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the values of \(x\) that satisfy the inequalities. $$ -6

Step-by-Step Solution

Answer

The values of \(x\) that satisfy the inequalities are \(-4 < x < 6\).

Step 1: Solve the first inequality (-6 < x - 2)

To solve the first inequality, add 2 to both sides of the inequality: \[-6 + 2 < x - 2 + 2\] This simplifies to: \[-4 < x\]

Step 2: Solve the second inequality (x - 2 < 4)

Now, let's solve the second inequality. Add 2 to both sides of the inequality: \[x - 2 + 2 < 4 + 2\] This simplifies to: \[x < 6\]

Step 3: Combine the solutions

Now that we have the solutions for both inequalities, we can combine the solutions and express it as a single inequality, showing the range of values for \(x\): \[-4 < x < 6\] So, the solution for the compound inequality is \(x\) that satisfy \(-4 < x < 6\).

Key Concepts

Compound Inequalities

When we talk about compound inequalities, we refer to two or more inequalities that are joined together by the words 'and' or 'or'. These phrases determine how the solution sets of the inequalities interact with each other. For the given problem, the compound inequality is written as

• \(-6 < x - 2\), and
• \(x - 2 < 4\)

This specific compound inequality uses an 'and' to join the two inequalities, meaning we’re looking for values of \(x\) that satisfy both conditions simultaneously. Think of it as finding the overlap where both conditions hold true at the same time. Compound inequalities are often used in problems where you want to define a range in which a variable can lie, which is helpful in real-world applications such as setting limits or constraints.

Solving Inequalities

Solving inequalities is very similar to solving equations, with one key distinction – direction matters! When working through inequalities, much like with equations, you perform operations to isolate the variable. In the exercise above, this is demonstrated by adding the same number to both sides of the inequalities to isolate \(x\).

• For \(-6 < x - 2\), we added 2 to each side to get \(-4 < x\).
• For \(x - 2 < 4\), we also added 2 to both sides to get \(x < 6\).

By doing these operations, we ensure that the direction of the inequality remains the same, as long as we’re not multiplying or dividing by a negative number. Remember, if you do multiply or divide both sides of an inequality by a negative number, the direction of the inequality signs flips.

Linear Inequalities

Linear inequalities involve any inequality that concerns linear expressions, meaning expressions in which the variable is raised to the first power. In this exercise, the inequalities given (\(-6 < x - 2 < 4\)) are linear because the variable \(x\) is not squared, cubed, or raised to any other power.The process of handling linear inequalities typically involves similar steps to those used in equations:

• Simplifying both sides if needed,
• Getting the variable on one side, especially if dealing with multiple terms involving the variable,
• Isolating the variable to find its range or solution.

Linear inequalities are useful for situations where we need to express conditions or constraints, for instance, ensuring that a certain quantity stays within a set range, much like the solution \(-4 < x < 6\) explains the range within which \(x\) lies.