Problem 32 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ -\frac{1}{2} \leq \frac{4-3 x}{5} \leq \frac{1}{4} $$

Step-by-Step Solution

Answer

The solution is \([11/12, 13/6]\). Graph the shaded region between 11/12 and 13/6 with closed circles at each endpoint.

Step 1: Understand the Inequality Structure

The inequality is a compound inequality, which means it has two parts: \(-\frac{1}{2} \leq \frac{4-3x}{5}\) and \(\frac{4-3x}{5} \leq \frac{1}{4}\). We will solve each part separately and then find the intersection of their solutions.

Step 2: Solve the First Inequality

Solve \(-\frac{1}{2} \leq \frac{4-3x}{5}\).- Multiply both sides by 5 to eliminate the denominator: \(-5/2 \leq 4 - 3x\).- Subtract 4 from both sides: \(-5/2 - 4 \leq -3x\) \(-13/2 \leq -3x\).- Divide both sides by -3, remembering to flip the inequality symbol: \(13/6 \geq x\).- This can be rewritten as \(x \leq 13/6\).

Step 3: Solve the Second Inequality

Solve \(\frac{4-3x}{5} \leq \frac{1}{4}\).- Multiply both sides by 5: \(4 - 3x \leq 5/4\).- Subtract 4 from both sides: \(-3x \leq 5/4 - 4\) \(-3x \leq -11/4\).- Divide both sides by -3 and flip the inequality symbol: \(x \geq 11/12\).

Step 4: Find the Intersection

The solutions we have are: \(x \leq 13/6\) and \(x \geq 11/12\). The intersection of these inequalities is:\(11/12 \leq x \leq 13/6\).

Step 5: Express in Interval Notation

Convert the solution into interval notation.The solution in interval notation is \([11/12, 13/6]\).

Step 6: Graph the Solution Set

To graph the solution set, draw a number line.Mark and close circles at \(11/12\) and \(13/6\) to indicate that both endpoints are included.Shade the region on the number line between 11/12 and 13/6.

Key Concepts

Compound Inequality

When dealing with compound inequalities, you're handling a mathematical expression with two combined inequalities that share the same variable. In the given problem, we have \(-\frac{1}{2} \leq \frac{4-3x}{5}\) and \(\frac{4-3x}{5} \leq \frac{1}{4}\). These statements imply that the value \(\frac{4-3x}{5}\) must satisfy both conditions simultaneously. To solve a compound inequality, focus on treating and solving each inequality separately. You'll then combine their solutions to discover any values of \(x\) that make both inequalities true at the same time.

Interval Notation

Interval notation is a method for representing a set of numbers between two endpoints on a number line. It provides a concise way to express solutions to inequalities. When using interval notation, you’ll often use brackets. A square bracket \([\) or \(]\) indicates that the endpoint is included in the set, while parentheses \((\) or \()\) mean it's not included. Given the intersection \(11/12 \leq x \leq 13/6\), the interval notation becomes \([11/12, 13/6]\), clearly indicating that \(x\) includes both 11/12 and 13/6.

Solving Inequalities

Inequalities are solved using similar techniques as equations, but with additional rules to keep in mind. One critical rule is if you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. To solve \(-\frac{1}{2} \leq \frac{4-3x}{5}\), follow these steps:

• Multiply each side by 5 to remove the denominator: \(-5/2 \leq 4 - 3x\).
• Subtract 4 from both sides: \(-13/2 \leq -3x\).
• Divide each side by -3 and flip the inequality: \(x \leq 13/6\).

For the second inequality \(\frac{4-3x}{5} \leq \frac{1}{4}\), the steps are:

• Multiply through by 5: \(4 - 3x \leq 5/4\).
• Subtract 4: \(-3x \leq -11/4\).
• Divide by -3 and flip the inequality: \(x \geq 11/12\).

In each case, careful attention to these rules ensures accurate solutions.

Inequality Graphing

Graphing inequalities visually represents the solution set on a number line. It’s a direct way to see which values satisfy the inequality. For our solution \(11/12 \leq x \leq 13/6\), graphing involves:

• Drawing a horizontal number line.
• Marking the numbers 11/12 and 13/6.
• Using closed circles on both numbers indicating these values themselves are included.
• Shading the section of the line between these two points to show all numbers \(x\) within this range satisfy the inequalities.

The closed circles confirm border values satisfy the inequality, illustrating the complete solution range.