Problem 34 - Algebra and Trigonometry

Question

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 in interval notation is \([\frac{11}{12}, \frac{13}{6}]\).

Step 1: Understanding the Compound Inequality

The given problem presents a compound inequality: \[-\frac{1}{2} \leq \frac{4-3x}{5} \leq \frac{1}{4}\] This means we need to solve for \(x\) while ensuring it satisfies both inequalities concurrently, i.e., \[-\frac{1}{2} \leq \frac{4-3x}{5}\] and \[\frac{4-3x}{5} \leq \frac{1}{4}\].

Step 2: Solving the Left Part of the Inequality

First, solve \[-\frac{1}{2} \leq \frac{4-3x}{5}.\] Multiply every term by 5 to eliminate the fraction: \[-\frac{5}{2} \leq 4 - 3x.\] Subtract 4 from each side: \[-\frac{5}{2} - 4 \leq -3x.\] Convert \(-4\) to a fraction with denominator 2: \[-\frac{8}{2} = -4.\] So, \[-\frac{5}{2} - \frac{8}{2} = -\frac{13}{2} \leq -3x.\] Divide every term by \(-3\), remembering to flip the inequality sign: \[\frac{13}{6} \geq x.\] This can be written as \[x \leq \frac{13}{6}.\]

Step 3: Solving the Right Part of the Inequality

Now solve \[\frac{4-3x}{5} \leq \frac{1}{4}.\] Multiply every term by 5: \[4 - 3x \leq \frac{5}{4}.\] Subtract 4 from each side: \[-3x \leq \frac{5}{4} - 4.\] Convert \(4\) into a fraction with denominator 4: \[-3x \leq \frac{5}{4} - \frac{16}{4}.\] Calculate: \[-3x \leq -\frac{11}{4}.\] Divide each term by \(-3\), flipping the inequality sign: \[x \geq \frac{11}{12}.\]

Step 4: Combining Solutions and Expressing in Interval Notation

Combine the solutions for \(x\):From Step 2, \[x \leq \frac{13}{6}\]From Step 3, \[x \geq \frac{11}{12}\]Thus, the intersection (solution set) is \[\frac{11}{12} \leq x \leq \frac{13}{6}\].In interval notation, this is \[\left[ \frac{11}{12}, \frac{13}{6}\right].\]

Step 5: Graphing the Solution Set

On a number line, mark \(\frac{11}{12}\) and \(\frac{13}{6}\) and shade the region between them, including the endpoints. This represents all numbers \(x\) such that \[\frac{11}{12} \leq x \leq \frac{13}{6}.\]

Key Concepts

Compound Inequalities

Compound inequalities are a type of problem where two individual inequalities are combined into one statement. Typically, they are expressed with the word "and," meaning both conditions must be true simultaneously. They might also be expressed with "or," indicating that at least one condition can be true for the compound inequality to hold as a solution.
When dealing with a compound inequality "and," as in the case of the problem \[-\frac{1}{2} \leq \frac{4-3x}{5} \leq \frac{1}{4}\], we have to resolve two separate inequalities and ensure that the solutions meet both conditions. To manage this, you treat the inequality as two separate expressions:

• \(-\frac{1}{2} \leq \frac{4-3x}{5}\)
• \(\frac{4-3x}{5} \leq \frac{1}{4}\)

This means finding a common solution that satisfies both.
Compound inequalities are essential in determining ranges of possible solutions that do not simply drift towards infinity. By solving each part independently and intersecting results, you can find the actual set of solutions.

Interval Notation

Once you have the solution for the inequality, expressing it in interval notation is a neat way to communicate which values work. Interval notation provides a clear visual on the range of solutions, showing endpoints and whether they are included or excluded.
In our example, the solution we arrived at was:\[\frac{11}{12} \leq x \leq \frac{13}{6}\].
In interval notation, we represent this range by using brackets and parentheses:

• The square bracket \([\) or \(]\) shows that an endpoint is included (meaning \(x\) can be equal to this number).
• A parenthesis \(()\) or \(()\) signifies that the endpoint is not included.

Thus, our solution becomes \[\left[ \frac{11}{12}, \frac{13}{6} \right]\], showcasing that all numbers between and including \(\frac{11}{12}\) and \(\frac{13}{6}\) are part of the solution.
This succinct form not only saves space but also provides clarity on the "interval" (range) of valid solutions quickly.

Inequality Graphing

Graphing a compound inequality like \[-\frac{1}{2} \leq \frac{4-3x}{5} \leq \frac{1}{4}\] provides a visual representation of all the values that satisfy the inequality. This is particularly useful for seeing the solution set clearly.
To graph the inequality on a number line:

• Start by marking the endpoints of the solution range, which we've determined are \(\frac{11}{12}\) and \(\frac{13}{6}\).
• Using closed circles or points, indicate these values on the number line because both endpoints are included in the solution set.
• Shade the region between these endpoints, illustrating that all the numbers in this segment satisfy the compound inequality.

Graphing helps to visually affirm the solution and serves educational purposes by showing how solutions emerge from algebraic manipulation.
It also aids in understanding the intersectionality of solutions in compound inequalities, as only the region satisfying both parts should be shaded.