Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ -4<3 x+2<5 $$

Step-by-Step Solution

Answer

The solution set is \((-2, 1)\) and is represented as a shaded region between -2 and 1 on a number line.

Step 1: Break Down the Compound Inequality

The given inequality is \[ -4 < 3x + 2 < 5 \] This is a compound inequality that can be separated into two inequalities:1. \( -4 < 3x + 2 \)2. \( 3x + 2 < 5 \)

Step 2: Solve the First Inequality

Let's solve \( -4 < 3x + 2 \). Subtract 2 from both sides:\[ -4 - 2 < 3x \] Simplify:\[ -6 < 3x \] Now, divide both sides by 3:\[ -2 < x \] Thus, this inequality is \( x > -2 \).

Step 3: Solve the Second Inequality

Now, solve \( 3x + 2 < 5 \). Subtract 2 from both sides:\[ 3x < 5 - 2 \]Simplify:\[ 3x < 3 \] Divide both sides by 3:\[ x < 1 \] Thus, this inequality is \( x < 1 \).

Step 4: Combine the Solutions

Now, combine the solutions from the previous steps. Our solution is where:\[ -2 < x < 1 \]This means that \( x \) must be greater than \(-2\) and less than \(1\).

Step 5: Express the Solution in Interval Notation

The interval notation for \( -2 < x < 1 \) is:\( (-2, 1) \)This denotes all the real numbers \( x \) that are greater than \(-2\) and less than \(1\).

Step 6: Graph the Solution

Sketch a number line. Plot an open circle at \( -2 \) and \( 1 \) to indicate that these values are not included. Shade the region between these two points to represent all numbers \( x \) such that \( -2 < x < 1 \).

Key Concepts

Interval Notation

Interval notation is a mathematical method used to describe solutions for inequalities. It offers a concise way to represent the set of numbers that satisfy a particular inequality. Our compound inequality \( -2 < x < 1 \) can be expressed in interval notation as \( (-2, 1) \). This notation signifies that \( x \) is within the range of \(-2\) and \(1\), but does not include \(-2\) and \(1\) themselves.

• The parenthesis \( ( \) and \( ) \) indicate that the endpoints are not included in the set.
• In contrast, square brackets \( [ \) and \( ] \) would be used if the endpoints were part of the solution set (i.e., \( [a, b] \) includes \(a\) and \(b\)).
• It is essential to always write the smaller number first followed by the larger number, keeping in mind that the interval is written from left-to-right on the number line.

Interval notation is particularly useful because it provides a clear, unambiguous description of the set of possible values, which is essential for communication in mathematics.

Compound Inequalities

Compound inequalities are simply a combination of two or more inequalities connected by the words “and” or “or”. They describe a range of possible solutions making them versatile tools in solving problems involving multiple conditions. In our example, the inequality \( -4 < 3x + 2 < 5 \) is a compound inequality.

• This means we have two separate inequalities within one expression:
• First: \( -4 < 3x + 2 \)
• Second: \( 3x + 2 < 5 \)
• They must both be true at the same time.

The goal is to find a solution range where both conditions are met. After solving the two inequalities separately, we combine the results to create a single solution set.

Number Line Graphing

Graphing on a number line is a visual way to represent the solutions of an inequality. It helps to quickly convey the range of solutions and whether endpoints are included. For the inequality \( -2 < x < 1 \) we plot it on a number line as follows:

• Draw a horizontal line that represents a span of numbers relevant to the inequality.
• Mark open circles at the points corresponding to \(-2\) and \(1\). These circles indicate that these numbers are not part of the solution.
• Shade the region between \(-2\) and \(1\) to denote the set of all numbers that satisfy \( -2 < x < 1 \).

This graphical method reinforces the solution's clarity, providing an immediate visual sense that aids understanding, especially for those who are more visually inclined.

Algebraic Manipulation

Algebraic manipulation is a core skill when solving inequalities. It involves rearranging equations to isolate the variable of interest. Understanding the basic rules of algebra helps in breaking down complex inequalities into simpler parts. Here’s how we tackled the compound inequality \( -4 < 3x + 2 < 5 \):

• First, we handled each inequality separately.
• For \( -4 < 3x + 2 \), we subtracted \(2\) from both sides to simplify it and then divided by \(3\):
  \( -4 - 2 < 3x \) leading to \( -6 < 3x \) and eventually \( -2 < x \).
• For \( 3x + 2 < 5 \), we also subtracted \(2\), then divided to isolate \(x\):
  \( 3x < 3 \) simplifying it to \( x < 1 \).

Effective algebraic manipulation hinges on maintaining the inequality’s balance by performing the same operations on both sides. This ensures the inequality remains true. It is important to remember that if you multiply or divide by a negative number, the inequality sign must be flipped.