Problem 32 - Algebra and Trigonometry

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ -3 \leq 3 x+7 \leq \frac{1}{2} $$

Step-by-Step Solution

Answer

The solution is \([-\frac{10}{3}, -\frac{13}{6}]\). Graph as a line with endpoints at \(-\frac{10}{3}\) and \(-\frac{13}{6}\).

Step 1: Break Down the Compound Inequality

The given compound inequality is: \[-3 \leq 3x + 7 \leq \frac{1}{2}\]This can be separated into two inequalities:1. \(-3 \leq 3x + 7\)2. \(3x + 7 \leq \frac{1}{2}\)

Step 2: Solve First Inequality

Let's solve the first inequality:\(-3 \leq 3x + 7\)Subtract 7 from both sides to isolate the term with \(x\): \[-3 - 7 \leq 3x\]Simplifying gives:\[-10 \leq 3x\]Divide both sides by 3 to solve for \(x\):\[x \geq -\frac{10}{3}\]

Step 3: Solve Second Inequality

Let's solve the second inequality:\(3x + 7 \leq \frac{1}{2}\)Subtract 7 from both sides to isolate the term with \(x\):\[3x \leq \frac{1}{2} - 7\]Simplifying gives:\[3x \leq -\frac{13}{2}\]Divide both sides by 3 to solve for \(x\):\[x \leq -\frac{13}{6}\]

Step 4: Combine Solutions

The solutions from the two inequalities are:- From the first inequality: \(x \geq -\frac{10}{3}\)- From the second inequality: \(x \leq -\frac{13}{6}\)To find the combined solution, notice you are finding a range where both conditions are valid. Thus, the solution is:\[-\frac{10}{3} \leq x \leq -\frac{13}{6}\]

Step 5: Express Solution in Interval Notation

Write the solution in interval notation:\[-\frac{10}{3}, -\frac{13}{6}\]

Step 6: Graph the Solution

To graph the solution on a number line, draw a line from \(x = -\frac{10}{3}\) to \(x = -\frac{13}{6}\), and draw solid circles at both \(-\frac{10}{3}\) and \(-\frac{13}{6}\) to indicate that these values are included in the solution.

Key Concepts

Interval Notation

Interval notation is a concise way of expressing a range of numbers, especially when dealing with solutions of inequalities. It uses parentheses \(\) or brackets \[\] to show which numbers are included or excluded in a solution set.
There are a few key points to remember about interval notation:

• Brackets \[a, b\] are used when numbers \(a\) and \(b\) are included in the solution. This means that both endpoints are part of the interval.
• Parentheses \(a, b\) exclude the numbers \(a\) and \(b\) from the set, indicating that the endpoints are not part of the interval.
• A combination \[a, b\) or \(a, b\] includes one endpoint and excludes the other.

In this exercise, the solution set from the linear inequality \(-\frac{10}{3}\leq x \leq -\frac{13}{6}\) is expressed in interval notation as \([-\frac{10}{3}, -\frac{13}{6}]\). The brackets indicate that both \(-\frac{10}{3}\) and \(-\frac{13}{6}\) are included in the interval, marking all values \(x\) in between as part of the solution.

Compound Inequality

A compound inequality involves two separate inequalities that are combined into one statement using the words 'and' or 'or'. The combined statement defines a set of solutions that satisfies both inequalities.
In mathematical terms:

• The "and" compound inequality means that all conditions in the inequalities must be satisfied simultaneously.
• The "or" compound inequality means that at least one condition in the inequalities must be satisfied.

For the inequality \(-3 \leq 3x + 7 \leq \frac{1}{2}\), we are looking at an "and" situation. Both individual conditions \(-3 \leq 3x + 7\) and \(3x + 7 \leq \frac{1}{2}\) must be true at the same time.
To solve compound inequalities, we can treat them as two separate equations initially, as shown in the step-by-step solution. After solving each part, you "combine" the solutions to find the overlapping range that fulfills both conditions. This combined solution is then expressed using interval notation as \([-\frac{10}{3}, -\frac{13}{6}]\).

Number Line Representation

Representing solutions on a number line helps visualize the range of numbers included in an inequality or compound inequality solution. Here's how you can accurately represent the interval \([-\frac{10}{3}, -\frac{13}{6}]\) on a number line:

• Start by marking the numbers \(-\frac{10}{3}\) and \(-\frac{13}{6}\) on the number line as reference points. These numbers represent the endpoints of your interval.
• Use solid circles or dots to indicate these endpoints on the number line. The solid markers show that \(-\frac{10}{3}\) and \(-\frac{13}{6}\) are included in the solution set.
• Draw a line segment connecting the solid circles. This segment represents all real numbers \(x\) that fall within this interval, clearly showing you the range of values that solve the inequality.

The number line is a valuable tool for visual learners. It graphically illustrates the scope of solutions, confirming the combined interval solution set of \([-\frac{10}{3}, -\frac{13}{6}]\). By checking the graph, you ensure you understand that all values between \(-\frac{10}{3}\) and \(-\frac{13}{6}\), including the endpoints, meet the requirements of the original compound inequality statement.