Problem 52 - Algebra for College Students

Question

Solve each compound inequality using the compact form. Express the solution sets in interval notation. \(-2<3 x+4<2\)

Step-by-Step Solution

Answer

The solution set is \((-2, -\frac{2}{3})\).

Step 1: Break Down the Compound Inequality

We start with the compound inequality \(-2 < 3x + 4 < 2\). This is a single inequality with two parts which need to be solved simultaneously.

Step 2: Solve the Left Part of the Inequality

First, solve the left part of the inequality \(-2 < 3x + 4\). Subtract 4 from both sides: \(-2 - 4 < 3x\) which simplifies to \(-6 < 3x\). Then divide both sides by 3: \(-2 < x\).

Step 3: Solve the Right Part of the Inequality

Now solve the right part of the inequality \(3x + 4 < 2\). Subtract 4 from both sides to get \(3x < 2 - 4\) which simplifies to \(3x < -2\). Then divide both sides by 3: \(x < -\frac{2}{3}\).

Step 4: Combine the Inequalities

Join both results to establish a single inequality in compact form. We have \(-2 < x\) and \(x < -\frac{2}{3}\). Combine them: \(-2 < x < -\frac{2}{3}\).

Step 5: Express in Interval Notation

Translate the inequality \(-2 < x < -\frac{2}{3}\) into interval notation. Since the inequality is strict (no equality involved), we use parentheses: \((-2, -\frac{2}{3})\).

Key Concepts

Interval Notation

Interval notation is a mathematical expression used to denote the range of values that satisfy an inequality or a set of inequalities. Understanding interval notation is important because it allows for a more concise and universally understood representation of these solutions.
In interval notation:

• Parentheses, \((... )\), are used to represent a strict inequality, meaning the endpoints are not included in the set.
• Brackets, \[\[ \] \], are used when the inequality includes equality, indicating the endpoints are included.
• A comma is used to separate the lower and upper bounds of the interval.

For the exercise \(-2 < x < -\frac{2}{3}\), this was expressed in interval notation as \((-2, -\frac{2}{3})\), indicating all values \(x\) strictly between -2 and -\(\frac{2}{3}\) are part of the solution. Interval notation provides a quick snapshot of all possible solutions without listing each individually.

Solving Inequalities

Solving inequalities involves finding the range of values that make an inequality true. These can be linear or non-linear and may involve single-variable expressions. To solve a compound inequality, you must consider two or more inequalities at the same time and find values that satisfy all conditions.The steps generally involve:

• Isolating the variable on one side of the inequality.
• Performing arithmetic operations, like addition, subtraction, multiplication, or division, while remembering that multiplying or dividing by a negative number flips the inequality sign.
• Combining solutions when dealing with compound inequalities.

For example, in the compound inequality \(-2 < 3x + 4 < 2\), we isolated \(x\) by first solving each part separately:

• For the part \(-2 < 3x + 4\), we ultimately found \(-2 < x\).
• For \(3x + 4 < 2\), we determined \(x < -\frac{2}{3}\).
• These solutions then combined to give \(-2 < x < -\frac{2}{3}\).

This shows the process of solving multiple inequalities simultaneously to ensure all conditions are met.

Algebraic Expressions

Algebraic expressions form the basis of constructing and solving inequalities. These expressions consist of variables, numbers, and operations that define specific relationships or rules.Key features of algebraic expressions include:

• They may include terms, such as \(3x\) or constant terms like \(4\).
• Operations such as addition, subtraction, multiplication, and division link the terms together.
• They can be rearranged or simplified using algebraic rules to isolate variables and solve inequalities.

In the original problem, the algebraic expression involved was \(3x + 4\). Solving this led to isolating the variable \(x\) to find values within specific bounds. Understanding how to manipulate these expressions is crucial for working through more complicated inequality scenarios and can be applied to many mathematical and real-world problems.