Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 0 \leq 4-\frac{1}{3} x<2 $$

Step-by-Step Solution

Answer

\((6, 12]\)

Step 1: Analyze the Double Inequality

The given inequality is a compound inequality: \( 0 \leq 4 - \frac{1}{3}x < 2 \). It means that \( 4 - \frac{1}{3}x \) is greater than or equal to 0 and less than 2. We will break it into two separate inequalities and solve each one independently.

Step 2: Solve the Left Inequality

The left inequality from the compound inequality is \( 0 \leq 4 - \frac{1}{3}x \). Subtract 4 from both sides: \( -4 \leq -\frac{1}{3}x \). Then, multiply both sides by -3 (remember to flip the inequality sign, since we are multiplying by a negative number): \( 12 \geq x \). This simplifies to \( x \leq 12 \).

Step 3: Solve the Right Inequality

The right inequality from the compound inequality is \( 4 - \frac{1}{3}x < 2 \). Subtract 4 from both sides: \( -\frac{1}{3}x < -2 \). Then, multiply both sides by -3 (again flipping the inequality sign): \( x > 6 \).

Step 4: Combine the Solutions

To satisfy both inequalities, \( x \) must be greater than 6 and less than or equal to 12. Combining these inequalities, we get \( 6 < x \leq 12 \). This can be expressed in interval notation as \((6, 12]\).

Key Concepts

Compound Inequalities

Compound inequalities involve combining two or more inequalities that must be true at the same time. In the exercise, the compound inequality is given as:

• \(0 \leq 4 - \frac{1}{3}x < 2\)

This tells us two things about the variable \(x\):

• \(x\) must make \(4 - \frac{1}{3}x\) greater than or equal to 0.
• \(x\) must make \(4 - \frac{1}{3}x\) less than 2.

To solve a compound inequality, we separate these two conditions and solve them independently. The solutions to these individual inequalities are then combined to find the values of \(x\) that satisfy both conditions together. This combination is crucial because it forms a range of values that the variable can take.
Remember, when solving these inequalities, the rules applied to both inequalities are the same, but each must be handled separately to ensure accuracy when combining them later.

Interval Notation

Interval notation is a mathematical shorthand used to express a set of numbers along a continuous range. It is especially handy when discussing solutions to inequalities. In our solution, the interval notation \((6, 12]\) is used.

Here's a quick breakdown of the symbols:

• Parentheses \((\) or \()\) indicate that an endpoint is not included. For example, \(x > 6\) is expressed as \((6, \infty)\).
• Brackets \([\) or \(]\) show that an endpoint is included. For example, \(x \leq 12\) is expressed as \((-\infty, 12]\).

When reading the interval \((6, 12]\), it signifies that \(x\) is greater than 6 but less than or equal to 12. The parenthesis next to 6 means 6 is not part of the solution, while the bracket next to 12 means 12 is included.

Interval notation succinctly represents the solution space and is particularly useful for communicating the range over which an inequality holds true.

Mathematical Problem Solving

Solving mathematical problems, such as inequalities, often involves breaking down complex instructions into manageable steps. This approach improves understanding and ensures accuracy.

• First, carefully read and comprehend the problem statement, as seen with the given compound inequality: \(0 \leq 4 - \frac{1}{3}x < 2\).
• Break it into smaller, easier-to-solve parts. Tackle the left and right inequalities separately: find solutions for \(0 \leq 4 - \frac{1}{3}x\) and \(4 - \frac{1}{3}x < 2\).
• Solve each part using standard algebraic methods like addition, subtraction, multiplication, or division. Be aware of rules: multiplying or dividing by a negative number reverses the inequality sign.
• Combine your results logically, ensuring all conditions are satisfied simultaneously. Check results for accuracy.

Problem-solving in mathematics requires patience, precision, and practice. By breaking tasks into smaller steps, one can simplify seemingly daunting tasks and arrive at the correct solution efficiently.