Problem 32 - Precalculus Mathematics for Calculus

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 \([-\frac{10}{3}, -\frac{13}{6}]\) and is graphed with a shaded segment between these points.

Step 1: Break Down the Compound Inequality

The given inequality is a compound inequality: \[-3 \leq 3x + 7 \leq \frac{1}{2}\] We need to solve this as two separate inequalities: 1. \(-3 \leq 3x + 7\)2. \(3x + 7 \leq \frac{1}{2}\).

Step 2: Solve the First Inequality

Solve the inequality \(-3 \leq 3x + 7\):Subtract 7 from both sides:\[-3 - 7 \leq 3x\]\[-10 \leq 3x\]Divide both sides by 3:\[-\frac{10}{3} \leq x\].

Step 3: Solve the Second Inequality

Solve the inequality \(3x + 7 \leq \frac{1}{2}\):Subtract 7 from both sides:\[3x \leq \frac{1}{2} - 7\]\[3x \leq \frac{1}{2} - \frac{14}{2}\]\[3x \leq -\frac{13}{2}\]Divide both sides by 3:\[x \leq -\frac{13}{6}\].

Step 4: Combine the Solutions

Combine the results from Steps 2 and 3:\[-\frac{10}{3} \leq x \leq -\frac{13}{6}\].

Step 5: Express the Solution in Interval Notation

The solution in interval notation is:\[\left[-\frac{10}{3}, -\frac{13}{6}\right]\]

Step 6: Graph the Solution Set

Draw a number line. Shade the region between \(-\frac{10}{3}\) and \(-\frac{13}{6}\). Both endpoints are included, so represent them with solid dots.

Key Concepts

Compound Inequalities

Compound inequalities are combinations of two individual inequalities joined by the word "and" or "or." These inequalities describe a range of values that satisfy both conditions. In our problem, we have a compound inequality: \[-3 \leq 3x + 7 \leq \frac{1}{2}\].This can be split into two separate inequalities:

• \(-3 \leq 3x + 7\)
• \(3x + 7 \leq \frac{1}{2}\)

Each inequality needs to be solved separately to find the range of possible values for \(x\). After solving each one, the intersection of the solutions yields the answer to the compound inequality. Always remember:

• "And" indicates that solutions must satisfy both inequalities simultaneously.
• "Or" would mean solutions can satisfy either of the inequalities independently.

Compound inequalities are useful for expressing constraints in real-world problems, where multiple conditions must be met at the same time.

Interval Notation

Interval notation is a system used to represent a range of numbers along a number line. It is concise and easy to understand. After solving a compound inequality, you express the solution in interval notation. For our inequality, \[-\frac{10}{3} \leq x \leq -\frac{13}{6}\],we need to represent this solution using interval notation. In interval notation:

• Square brackets \([\,]\) denote that an endpoint is included in the solution (i.e., the endpoint satisfies the inequality).
• Parentheses \((\,)\) would denote that an endpoint is not included.

Therefore, the interval notation for the solution is \(\left[-\frac{10}{3}, -\frac{13}{6}\right]\), indicating that \(-\frac{10}{3}\) and \(-\frac{13}{6}\) are both part of the solution set.

Graphing Inequalities

Graphing inequalities is a visual method to represent the solutions of an inequality on a number line. This not only helps in understanding the solution but also provides a quick way to verify our work. Once we have the solution in interval notation, \(\left[-\frac{10}{3}, -\frac{13}{6}\right]\),we can draw it on a number line:

• Draw a horizontal line representing the number line.
• Mark the points \(-\frac{10}{3}\) and \(-\frac{13}{6}\) on this line.
• Since both endpoints are included in the solution, use solid dots to represent them.
• Shade the region between \(-\frac{10}{3}\) and \(-\frac{13}{6}\) to show all the possible values \(x\) can take.

Graphing provides an intuitive picture of the solution set, showing all numbers that satisfy the original inequality clearly and effectively.