Problem 19 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the values of \(x\) that satisfy the inequalities. $$ x+3>1 \text { and } x-2<1 $$

Step-by-Step Solution

Answer

The values of \(x\) that satisfy both inequalities are given by \(-2 < x < 3\).

Step 1: Solve the first inequality

We need to solve the inequality \(x + 3 > 1\). To do this, we subtract 3 from both sides of the inequality: \(x + 3 - 3 > 1 - 3\) Which simplifies to: \(x > -2\)

Step 2: Solve the second inequality

Now we need to solve the inequality \(x - 2 < 1\). To do this, we add 2 to both sides of the inequality: \(x - 2 + 2 < 1 + 2\) Which simplifies to: \(x < 3\)

Step 3: Combine the solution sets

We have found that \(x > -2\) and \(x < 3\). To find the values of \(x\) that satisfy both inequalities, we need to find the intersection of these two solution sets. In other words, we need to find the values of \(x\) that are greater than -2 and less than 3 at the same time. We can represent this as: \(-2 < x < 3\) The solution is the range of values for \(x\) between -2 and 3, excluding -2 and 3 themselves.

Key Concepts

Solving Inequalities

Solving inequalities involves determining the values that satisfy the inequality condition. It is a bit different from solving equations, where you find exact values for the variable. Here, instead of a single solution, you might find a range or set of values. To solve an inequality like \(x + 3 > 1\), you perform operations to isolate the variable, much like you do in equations. You can add, subtract, multiply, or divide both sides, keeping the inequality sign consistent, except when multiplying or dividing by a negative number, which flips the sign.

• Example: To solve \(x + 3 > 1\), subtract 3 from both sides to get \(x > -2\).
• Remember: Keep the variable on one side to easily determine the range or set of solutions.

Once solved, inequalities tell us what values make the inequality true, expressed as a range or interval.

Intersection of Solution Sets

When dealing with more than one inequality, finding the solution that satisfies all conditions involves finding the intersection of their solution sets. This means identifying the common values between the solutions of each inequality.

• For \(x + 3 > 1\), the solution set is \(x > -2\).
• For \(x - 2 < 1\), the solution set is \(x < 3\).

The intersection occurs where these solutions overlap, resulting in a combined range:

• In this case, \(-2 < x < 3\).

Understanding intersection helps solve systems of inequalities, revealing the common, shared values that fit all given conditions.

Mathematical Notation

Mathematical notation is a language of symbols used to define and work with numbers and operations. In inequalities, symbols play a crucial role:

• "\(>\)" and "\(<\)" denote strict inequalities, meaning the number cannot equal the boundary value.
• "\( \ge \)" and "\( \le \)" include equality with the number on the boundary.

For example, in the solution \(-2 < x < 3\), the notation indicates that \(x\) is greater than \(-2\) but less than \(3\), excluding both boundary values. It's essential to understand these notations to interpret mathematical solutions accurately.

• This notation helps visualize solutions as continuous ranges on the number line.

Practicing with these symbols can improve problem-solving speed and accuracy.