Problem 20 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the values of \(x\) that satisfy the inequalities. $$ x-4 \leq 1 \text { and } x+3>2 $$

Step-by-Step Solution

Answer

The short answer to the given inequality system is: \(-1 < x \leq 5\), which represents all values of x that satisfy both inequalities.

Step 1: Isolate x in the first inequality

Add 4 to both sides of the inequality \(x - 4 \leq 1\) to isolate x: \[ x - 4 + 4 \leq 1 + 4 \]

Step 2: Simplify the first inequality

Combine the numbers on both sides of the inequality: \[ x \leq 5 \] The solution to the first inequality is any value of x that is less than or equal to 5. ##Solving Inequality 2##

Step 3: Isolate x in the second inequality

Subtract 3 from both sides of the inequality \(x + 3 > 2\) to isolate x: \[ x + 3 - 3 > 2 - 3 \]

Step 4: Simplify the second inequality

Combine the numbers on both sides of the inequality: \[ x > -1 \] The solution to the second inequality is any value of x that is greater than -1. ##Combining the Solution##

Step 5: Determine the intersection of inequality solutions

Since we need to find the values of x that satisfy both inequalities, we are looking for the intersection of the two solutions. In general terms, a value of x must satisfy both \(x \leq 5\) and \(x > -1\).

Step 6: Write the final solution

To satisfy both inequalities, x must be within the following range: \[ -1 < x \leq 5 \] This range is the solution to the given inequality system that represents all values of x that satisfy both inequalities.

Key Concepts

Understanding Solving Inequalities

Solving inequalities is a foundational skill in algebra. It involves finding the range of values for a variable that makes the inequality statement true. Unlike equations which have a single or a finite number of solutions, inequalities can have an infinite set of solutions.
Understanding the method to solve inequalities helps in everyday decision-making scenarios where precise outcomes are not always available. For example, in budgeting or planning, one might need to know the maximum or minimum values that satisfy a certain requirement.
To solve an inequality, follow these steps:

• Identify the terms involving the variable.
• Perform operations (addition, subtraction, multiplication, division) on both sides of the inequality to isolate the variable.
• Simplify the expression to express the inequality in its simplest form.

Keep in mind that if you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must change. This rule is crucial as it ensures the inequality maintains its truth.

Understanding the Intersection of Solutions

When dealing with two or more inequalities, the goal is to find a solution that satisfies all conditions simultaneously. This is known as the intersection of solutions. For example, if one inequality states that a variable must be less than or equal to 5 and another states that it must be greater than -1, the combined solution will lie between those two points.
Determining the intersection involves:

• Solving each inequality separately to find individual solution sets.
• Identifying the overlapping values or common range from these sets.

Intersection is visually easy to understand on a number line where the solutions of each inequality can be marked, and the overlapping section represents the intersection. This visualization technique aids in grasping the concept quickly.

Approaching Mathematical Problem Solving

Mathematical problem solving doesn't just involve calculations; it requires a strategic approach. When tackling problems like inequalities, a clear methodical approach can simplify the process.
Begin by reading the problem carefully to understand the requirements. Break it down into manageable steps and tackle each part systematically. Focus on:

• Identifying key information and constraints.
• Choosing the right methods and operations to simplify the problem.
• Keeping the solution paths well-organized to track your math work easily.

Finally, after solving, it's essential to evaluate if the solution fits within the problem’s context. Double-checking your work ensures accuracy and helps reinforce learning, which builds confidence in solving future problems.