Problem 33 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 4>\frac{2-3 x}{7} \geq-2 $$

Step-by-Step Solution

Answer

The solution is \( \left( -\frac{26}{3}, \frac{16}{3} \right] \).

Step 1: Break down the double inequality

The inequality given is \( 4 > \frac{2 - 3x}{7} \geq -2 \). Let's first break it down into two separate inequalities: \( 4 > \frac{2 - 3x}{7} \) and \( \frac{2 - 3x}{7} \geq -2 \). We will solve each part separately.

Step 2: Solve the first inequality

Solve the first inequality: \( 4 > \frac{2 - 3x}{7} \). Multiply through by 7 to eliminate the fraction: \( 28 > 2 - 3x \). Rearrange to find \(-3x < 26 \) by subtracting 2 from both sides. Now, divide each side by -3, remembering to flip the inequality sign: \( x > -\frac{26}{3} \).

Step 3: Solve the second inequality

Now solve the second inequality: \( \frac{2 - 3x}{7} \geq -2 \). Again, multiply through by 7: \( 2 - 3x \geq -14 \). Rearrange to find \(-3x \geq -16 \) by subtracting 2 from each side. Divide each side by -3, remembering to flip the inequality sign: \( x \leq \frac{16}{3} \).

Step 4: Combine the solutions

Combine the solutions from Steps 2 and 3: \( x > -\frac{26}{3} \) and \( x \leq \frac{16}{3} \). The overlap of these solutions is an interval in which \( -\frac{26}{3} < x \leq \frac{16}{3} \).

Step 5: Express the solution in interval notation

The solution set can be written in interval notation: \( \left( -\frac{26}{3}, \frac{16}{3} \right] \). This interval represents all values of \( x \) that satisfy both parts of the original inequality.

Key Concepts

Interval Notation

Interval notation is a concise way of writing sets of numbers, often used to describe the solution sets of inequalities. It uses parentheses and brackets to show which numbers are included or excluded in a set. This notation is especially helpful when dealing with ranges of values.

For example, consider the interval \((-3, 5]\). The round parenthesis \(()\) at \(-3\) indicates that \(-3\) is not included in the interval. The square bracket \(()\) at \(5\) indicates that \(5\) is included.

• An open interval like \((-3, 5)\) means neither boundary is included.
• A closed interval like \([2, 6]\) means both endpoints are included.
• A half-open interval like \((3, 7]\) means \(3\) is not included, but \(7\) is.

In our example, the solution for the inequality is \((-\frac{26}{3}, \frac{16}{3}]\). This means all numbers \(x\) from slightly greater than \(-\frac{26}{3}\) to \(\frac{16}{3}\), including \(\frac{16}{3}\), satisfy the inequality.

Compound Inequalities

Compound inequalities involve two or more inequalities combined by the words 'and' or 'or'. In the exercise, we see a double inequality: \[4 > \frac{2-3x}{7} \geq -2\] which is a typical example of a compound inequality. These inequalities demand a solution that satisfies all parts of the system.

• 'And' compound inequalities: Require both conditions to be true simultaneously. This often results in an intersection of solutions.
• 'Or' compound inequalities: Require at least one condition to be true. This typically leads to a union of solutions.

In our problem, we dealt with an 'and' situation. We first broke down the compound inequality into simpler parts that were easier to handle individually. After solving each part, we combined them, ensuring the solution satisfied both conditions. Therefore, the final solution reflects numbers that fit into both sets of conditions.

Step-by-Step Solution

When faced with solving inequalities, a step-by-step approach breaks the process into manageable parts. This guided approach assists in keeping track of the problem and detecting errors before they compound.

Here’s a brief guide to the method:

• **Break the problem into parts**: As in our exercise, separate the compound inequality into two parts. This simplifies the problem.
• **Solve each part**: Figure out each inequality separately. This involves operations like multiplying, dividing, or rearranging terms. Be vigilant with inequalities: Remember to flip the sign when multiplying or dividing by a negative number.
• **Combine solutions**: Bring all parts back together and find the overlap, often visualized on a number line. Since we had an 'and' inequality, our solution is the intersection.
• **Express in interval notation**: Once satisfied with the solution, express it in the neat form of interval notation, like \((-\frac{26}{3}, \frac{16}{3}]\).

These steps, when followed meticulously, can simplify solving even complex inequality problems and sharpen algebraic skills over time.