Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

The values of x that satisfy the given inequalities are \(x < -1 \text{ or } x > 1\).

Step 1: Solve each inequality separately

We have two inequalities, let's solve them one by one. For inequality \(x + 1 > 2\): To isolate the x, subtract 1 from both sides of the inequality: \(x > 1\) For inequality \(x - 1 < -2\): To isolate the x, add 1 to both sides of the inequality: \(x < -1\)

Step 2: Combine the solutions

Since the original inequality has the "or" sign, any value of x that satisfies either of the inequalities will be a solution. For inequality \(x > 1\), the x values greater than 1 are the solutions. For inequality \(x < -1\), the x values less than -1 are the solutions. So the solution of the given compound inequality is: \(x < -1 \text{ or } x > 1\) The values of x that satisfy the given inequalities are the values less than -1 or greater than 1.

Key Concepts

Compound Inequalities

Compound inequalities are a special type of mathematical statement that involves more than one inequality being solved at the same time. Unlike a simple inequality, which involves only one statement, compound inequalities combine multiple inequalities using conjunctions like "and" or "or".

In a statement involving "and," both conditions must be true at the same time. Conversely, an "or" compound inequality requires that at least one of the conditions be true to satisfy the inequality.

It's useful to see compound inequalities as having a broader solution set since they provide multiple scenarios in which the statement holds true. Understanding compound inequalities is crucial when solving more complex mathematical problems, as they often form the backbone of mathematical reasoning.

Solving Inequalities

Solving inequalities is quite similar to solving equations, with some key differences. The goal is to isolate the variable, usually represented by an "x", on one side of the inequality sign (>, <, ≥, ≤).

To solve an inequality, you need to perform operations that maintain the inequality's truth – such as adding, subtracting, multiplying, or dividing both sides by the same number. However, a crucial point to remember is that when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.

In our example of the first inequality \(x+1 > 2\):

• Subtract 1 from both sides to isolate \(x\), yielding \(x>1\).

And for \(x-1 < -2\):

• Add 1 to both sides to get \(x < -1\).

These manipulations allow you to find the range of values for \(x\) that satisfy the conditions.

Mathematical Solutions

Mathematical solutions to inequalities reveal the set of possible values that satisfy the given statements. For compound inequalities that include "or," such as \(x+1 > 2\ or\ x-1 < -2\), the solution consists of values that satisfy either condition. This broadens the possible values that \(x\) can take.

For instance, if you solve \(x+1 > 2\), you find that \(x > 1\). On the other hand, solving \(x-1 < -2\) gives you \(x < -1\).

• The solution is all values of \(x\) such that \(x > 1\) or \(x < -1\).

These solutions are represented on a number line with two open ends, reflecting the infinite possibilities. Understanding these solutions can help students recognize the dynamic range of possibilities that inequalities offer, expanding problem-solving skills and mathematical insight.