Problem 63 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ \left|\frac{2-3 x}{5}\right| \geq 2 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -\frac{8}{3}] \cup [4, \infty)\).

Step 1: Understand the Absolute Inequality

The inequality given is \( \left|\frac{2-3x}{5}\right| \geq 2 \). An absolute value inequality like \( |A| \geq B \) implies that \( A \geq B \) or \( A \leq -B \). We'll apply this property to explore two cases for our inequality.

Step 2: Solve the First Case

In the first case, we set the expression inside the absolute value greater than or equal to 2: \[ \frac{2-3x}{5} \geq 2 \]First, multiply both sides by 5 to eliminate the fraction:\[ 2-3x \geq 10 \]Next, subtract 2 from both sides:\[ -3x \geq 8 \]Then divide by -3, remembering to reverse the inequality:\[ x \leq -\frac{8}{3} \]

Step 3: Solve the Second Case

In the second case, set the expression inside the absolute value less than or equal to -2:\[ \frac{2-3x}{5} \leq -2 \]Multiply both sides by 5:\[ 2-3x \leq -10 \]Subtract 2 from both sides:\[ -3x \leq -12 \]Divide by -3, again reversing the inequality:\[ x \geq 4 \]

Step 4: Combine the Solutions

The solutions of the inequality include both cases we solved:For the first case, \( x \leq -\frac{8}{3} \), and for the second case, \( x \geq 4 \). Thus, the solution set can be expressed in interval notation as:\[ (-\infty, -\frac{8}{3}] \cup [4, \infty) \]

Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities can sometimes be confusing, but they can be simplified by breaking them into two separate cases. Absolute value, represented as \(|A|\), measures the distance of a number, \(A\), from zero, without considering direction.
For an inequality like \(|A| \geq B\), it means that \(A\) is at least \(B\) units away from zero. This implies two situations:

• \(A \geq B\)
• \(A \leq -B\)

Applying these rules, you can dissect an absolute value inequality into two linear inequalities, which can then be solved separately for the variable. This approach allows us to find all possible solutions that fulfill the inequality conditions.

Expressing Solutions Using Interval Notation

Once you have determined the solution of an inequality, showing it properly in interval notation helps others understand the range or set of values that satisfy it. Interval notation provides a compact and precise way to express inequalities.
In interval notation, a solution can look like this:

• \((a, b)\): all numbers between \(a\) and \(b\), not including \(a\) or \(b\)
• \([a, b]\): all numbers from \(a\) to \(b\), including both \(a\) and \(b\)
• \((-\infty, b]\): all numbers less than or equal to \(b\)
• \([a, \infty)\): all numbers greater than or equal to \(a\)

The solution to our original inequality was expressed as two separate intervals: \((-\infty, -\frac{8}{3}]\) and \([4, \infty)\). This combined in union, \(\cup\), shows the full solution set where either condition can be met.

Solving Inequalities

Solving inequalities is a key algebra skill, as it involves determining the range of values that make an inequality true. It's similar to solving equations, but with inequalities, particular care must be taken with operations that affect the inequality's direction.
Key steps when solving linear inequalities include:

• Isolate the variable on one side of the inequality.
• If multiplying or dividing by a negative number, reverse the inequality sign.
• Check your solution by testing values from each interval.

In solving \(|\frac{2-3x}{5}| \geq 2\), we isolated \(x\) using similar steps but considered two cases due to the absolute value. This allowed us to find two distinct solution ranges, which accurately depict where the inequality holds true. Always remember, a solution is fully valid only when it satisfies the initial inequality fully in its defined domain.