Problem 63 - Algebra 2

Question

Solve each inequality. \(-2<3 x+1<7\)

Step-by-Step Solution

Answer

\(-1 < x < 2\).

Step 1: Understand the Compound Inequality

The inequality given is a compound inequality: \(-2 < 3x + 1 < 7\).This means that \(3x + 1\) should be greater than \(-2\) and also less than \(7\). We will solve each part of the inequality separately.

Step 2: Solve the Left Part of the Inequality

First, solve \(-2 < 3x + 1\).Subtract \(1\) from both sides:\[-2 - 1 < 3x\]\[-3 < 3x\]Now, divide both sides by \(3\):\[\frac{-3}{3} < x\]\[-1 < x\]

Step 3: Solve the Right Part of the Inequality

Now solve the second part, \(3x + 1 < 7\).Subtract \(1\) from both sides:\[3x < 7 - 1\]\[3x < 6\]Then, divide both sides by \(3\):\[x < \frac{6}{3}\]\[x < 2\]

Step 4: Combine the Results

Combine the results from both parts of the inequality:From Step 2, we have \(-1 < x\).From Step 3, we have \(x < 2\).Therefore, combining these gives:\(-1 < x < 2\).

Key Concepts

Solving Inequalities

Solving inequalities is a fundamental concept in algebra that extends the idea of solving equations to expressions that involve inequality signs, such as "<", ">", "≤", and "≥". Unlike equalities, solutions to inequalities can include a range of values, rather than a single value. To solve inequalities, we use processes similar to solving equations but with special rules when dealing with multiplication or division by negative numbers.

When solving inequalities:

• Treat them like equations by performing operations on both sides.
• Always maintain the inequality's direction unless multiplying or dividing by a negative number.
• Pay attention to any restrictions the inequalities might impose.

Understanding how to properly manipulate inequalities is crucial, as they frequently appear in various forms, such as compound inequalities, where multiple inequalities are connected and must hold true simultaneously.

Inequality Solution Steps

To find the solution to an inequality or a compound inequality, follow a structured approach that ensures each part of the inequality is addressed. Let's take the example of the compound inequality \(-2 < 3x + 1 < 7\):

• Step 1: Break Down the Compound Inequality Start by understanding that a compound inequality combines two or more inequalities that must all be true. In our example, this means solving both \(-2 < 3x + 1\) and \(3x + 1 < 7\).
• Step 2: Solve Each Part Deal with each inequality separately:
  • For \(-2 < 3x + 1\): subtract 1 from both sides resulting in \(-3 < 3x\), then divide by 3 giving \(-1 < x\).
  • For \(3x + 1 < 7\): subtract 1 from both sides resulting in \(3x < 6\), then divide by 3 giving \(x < 2\).
• Step 3: Combine the Results After solving each part, combine the results to express the range of solutions, giving us the compound solution: \(-1 < x < 2\).

This systematic approach ensures that each part of the compound inequality is addressed accurately, providing a clear set of solutions.

Algebraic Expressions

Algebraic expressions form the basis of solving inequalities and equations. They consist of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division.

When working with algebraic expressions in inequalities:

• Identify terms: Understand what numbers and variables are present and how they interact.
• Simplify wherever possible: Combine like terms and follow order-of-operations rules to simplify the expressions.
• Isolate the variable: Rearrange the expression to solve for the variable of interest, which usually involves moving terms around by performing inverse operations.

Mastering algebraic expressions allows for confident manipulation of inequalities. In the exercise, the expression \(3x + 1\) was central, requiring operations like subtraction and division to isolate \(x\). This step-by-step simplification is essential for reaching correct solutions in both single and compound inequalities.