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Chapter 4: Problem 88

Solve each inequality. Write the solution set in interval notation and graph it. $$ -5<3 t+4 \leq 13 $$

Short Answer

Expert verified
The solution is \((-3, 3]\). Graphically, it's shaded between open \(-3\) and closed \(3\).

Step by step solution

01

Break Down Inequality

The inequality consists of two parts: 1. \(-5 < 3t + 4\) 2. \(3t + 4 \leq 13\) Our task is to solve these two inequalities separately, and then combine their solutions.
02

Solve the First Inequality

Start with \(-5 < 3t + 4\). Subtract 4 from both sides to isolate the term with \(t\):\(-5 - 4 < 3t\)\(-9 < 3t\). Now, divide both sides by 3 to solve for \(t\):\(-3 < t\), which can be rewritten as \(t > -3\).
03

Solve the Second Inequality

Now solve \(3t + 4 \leq 13\). Subtract 4 from both sides:\(3t \leq 13 - 4\)\(3t \leq 9\). Divide both sides by 3 to solve for \(t\):\(t \leq 3\).
04

Combine Solutions

Combine the results of both inequalities. From Step 2: \(t > -3\) From Step 3: \(t \leq 3\) The combined solution is \(-3 < t \leq 3\).
05

Write in Interval Notation

The solution in interval notation is \((-3, 3]\). This means \(t\) can be any number greater than -3 and up to, including 3.
06

Graph the Solution

To graph the solution, use a number line: - Draw an open circle at -3 to indicate that -3 is not included. - Draw a closed circle at 3 to indicate that 3 is included. - Shade the region between -3 and 3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Interval Notation
Interval notation is a way of expressing the solution to inequalities in a concise and efficient manner. It uses brackets and parentheses to show which numbers are included in the solution.
For instance, consider the interval notation \((-3, 3]\):
  • The round bracket ( or ) at -3 indicates an open interval, meaning -3 is not included in the solution.
  • The square bracket [ or ] at 3 suggests a closed interval, indicating that 3 is included.
This notation replaces the need to write separate expressions for each part of the solution, making it especially helpful when expressing compound inequalities such as in the example \(-3 < t \leq 3\).
Knowing how to interpret these notations is critical for understanding where solutions fall on the number line.
The Basics of Inequality Graphing
Graphing inequalities is a visual approach to understanding solution sets. To graph an inequality, you utilize a number line to illustrate which values satisfy the inequality.
Here's how to graph the solution set \((-3, 3]\) from our exercise:
  • Draw a number line with numbers around the range of interest.
  • Place an open circle on -3, signifying that -3 is not part of the solution. Open circles are used whenever the inequality is strict, like \(t > -3\).
  • Place a closed circle at 3, indicating that 3 is included in the solution. Closed circles correspond to "less than or equal to" inequalities, such as \(t \leq 3\).
  • Shade the region between the two circles to represent all numbers that satisfy the inequality \(-3 < t \leq 3\).
By graphing inequalities, you can more easily visualize which values are part of the solution, making it easier to work with solutions in various contexts.
Finding Inequality Solutions
Solving inequalities involves finding all possible values that satisfy the inequality condition. Unlike equations, inequalities express a range of possible solutions rather than a single answer.
To solve inequalities like \(-5 < 3t + 4 \leq 13\), follow these steps:
  • Break down the inequality into separate, manageable parts. Solve each inequality by isolating the variable, as seen with each step in our example: solving \(-5 < 3t + 4\) and \(3t + 4 \leq 13\) individually.
  • Simplify each part to get expressions like \(t > -3\) and \(t \leq 3\).
  • Combine your findings to get a complete solution for the original compound inequality, such as \(-3 < t \leq 3\).
This combination reflects the intersection of the solutions for each part of the inequality. Recognizing how to solve these can empower you to tackle more complex problems with ease.

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Most popular questions from this chapter

Can the system \(\left\\{\begin{array}{l}{2 x+3 y=13} \\ {7 x-3 y=-5}\end{array}\right.\) be solved more easily using \right. the elimination method or the substitution method? Explain.

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