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 it's represented by an open interval on a number line between \(-2\) and \(1\).

Step 1: Understand the Inequality

We are given a compound inequality: \(-4 < 3x + 2 < 5\). This means the expression \(3x + 2\) is greater than \(-4\) and less than \(5\). We need to solve for \(x\).

Step 2: Split the Inequality

To solve this compound inequality, we treat it as two separate inequalities: \(-4 < 3x + 2\) and \(3x + 2 < 5\).

Step 3: Solve the First Inequality

For \(-4 < 3x + 2\):1. Subtract \(2\) from both sides to isolate the term with \(x\): \(-4 - 2 < 3x \rightarrow -6 < 3x\).2. Divide both sides by \(3\): \(-2 < x\).

Step 4: Solve the Second Inequality

For \(3x + 2 < 5\):1. Subtract \(2\) from both sides: \(3x < 5 - 2 \rightarrow 3x < 3\).2. Divide both sides by \(3\): \(x < 1\).

Step 5: Combine the Results

Combine the results from Step 3 and Step 4 to express the solution:- From the first part, \(-2 < x\).- From the second part, \(x < 1\).- Therefore, the solution is \(-2 < x < 1\).

Step 6: Write in Interval Notation

The solution \(-2 < x < 1\) is the interval notation \((-2, 1)\), meaning all numbers between \(-2\) and \(1\), not including \(-2\) and \(1\) themselves.

Step 7: Sketch the Graph

On a number line, draw an open circle at \(-2\) and another at \(1\). Shade the region in between the circles to represent the interval \((-2, 1)\). This shows all values of \(x\) that satisfy the inequality.

Key Concepts

Compound Inequalities

A compound inequality is like having two mail packages instead of one. It contains two separate inequalities joined by either "and" or "or." In our exercise, the compound inequality was \( -4 < 3x + 2 < 5 \). This is an "and" situation, meaning we are looking for values of \( x \) that fit both restrictions simultaneously.

Start by breaking it down. Imagine two statements: \(-4 < 3x + 2\) and \(3x + 2 < 5\). You solve each part on its own to uncover the possible values for \( x \). When both parts are resolved, as in this example, \(-2 < x\) and \(x < 1\), you have found a range of numbers that satisfy the entire statement.

This type of problem is great for practicing problem-solving skills because it requires finding common ground in two different scenarios.

Interval Notation

Interval notation is like using a shortcut to show a list of all the possible solutions for \( x \). It's a streamlined way to express a range of numbers, especially useful when describing solutions of inequalities.

In our example, the final result of the compound inequality is the range \(-2 < x < 1\). In interval notation, this is written as \((-2, 1)\).

Some key things to remember:

• A parenthesis \((\) means the endpoint is not included, like an open circle in graphing.
• A square bracket \([\) means the endpoint is included, marked by a closed circle.

You will use parentheses here because the numbers -2 and 1 are not part of the solution themselves.

Interval notation helps when summarizing solutions so they are easy to read and understand.

Graphing Inequalities

Graphing inequalities is like turning a math problem into a picture. It helps visualize the range of solutions on a number line. In the exercise, after solving \(-2 < x < 1\), you draw this on a number line to represent which numbers work.

For our example:

• Place open circles at -2 and 1, indicating these numbers are not part of the solution but are limits.
• Shade the segment between the circles because all numbers in this space satisfy the inequality.

Remember, an open circle means "not included," and a closed circle would mean "included."

Graphing inequalities helps convert algebraic solutions into a visual format, making it easier to verify the correctness of solved inequalities.