Problem 65 - Algebra and Trigonometry with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The solution is \((-\infty, \frac{7}{4})\).

Step 1: Identify the Absolute Value Expression

The given inequality is \( \frac{3}{|5-2x|}<2 \). Here \(|5-2x|\) is the absolute value expression that we need to address. This means \(5-2x\) could be either positive or negative, so we'll consider both cases.

Step 2: Consider the Case when \(5-2x\) is Positive

Assume \(5-2x > 0\), which implies \(5 > 2x\) and hence \(x < \frac{5}{2}\). In this scenario, the inequality \( \frac{3}{5-2x} < 2 \) simplifies to \(3 < 2(5-2x)\).

Step 3: Solve the Simplified Inequality for Positive Case

Solve \(3 < 2(5-2x)\):\[ 3 < 10 - 4x \]Subtraction gives \[ -7 < -4x \]Dividing by -4 (and flipping the inequality sign) yields \(x < \frac{7}{4}\). Thus, \(x < \min \left(\frac{5}{2}, \frac{7}{4}\right)\).

Step 4: Consider the Case when \(5-2x\) is Negative

Assume \(5-2x < 0\), meaning \(x > \frac{5}{2}\). The inequality becomes \(-\frac{3}{5-2x} < 2 \), which simplifies to solving \(-3 < 2x-5\).

Step 5: Solve the Simplified Inequality for Negative Case

Solve \(-3 < 2x-5\):\[ -3 + 5 < 2x \]Simplify to get \[ 2 < 2x \]Divide both sides by 2 to find \(x \):\[ x > 1 \]. But this solution is already fulfilled because in this case, we initially assumed \(x > \frac{5}{2}\).

Step 6: Combine Conditions and Express Solution in Interval Notation

From the positive case, we have \(x < \frac{7}{4} \) and \(x < \frac{5}{2}\), which simplifies to \(x < \frac{7}{4}\). Therefore, the overall solution in interval notation is \((-\infty, \frac{7}{4})\).

Key Concepts

Absolute Value

The absolute value represents the distance of a number from zero on the number line. It is an important concept in many mathematical operations because it ignores any negative signs, focusing only on the magnitude. In our inequality, \(\frac{3}{|5-2x|}<2\), the expression inside the absolute value, \(|5-2x|\), could potentially be either positive or negative. This absolute value influences how we evaluate and solve the inequality. When you see \(|5-2x|\), you should understand that you need to consider both possibilities for the expression, \(5-2x\) being either positive or negative. When \(5-2x > 0\), it remains as \(5-2x\); however, when \(5-2x < 0\), it becomes \(-(5-2x)\). This dual consideration is essential when solving equations involving absolute values.

Interval Notation

Once you solve an inequality, it's important to express the solution using interval notation. This notation is a way of describing a set of numbers on a number line. It uses brackets and parentheses to show which values are included or excluded from the set. For our exercise, we determined that \(x < \frac{7}{4}\). In interval notation, this is represented as \((-\infty, \frac{7}{4})\). The parenthesis next to \(\frac{7}{4}\) means that the endpoint \(\frac{7}{4}\) is not included in the solution set. The use of \(\infty\) signifies that the interval extends indefinitely in the negative direction. Interval notation offers a concise and clear way to represent solutions, especially useful when presenting solutions to inequalities.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying mathematical expressions in order to solve equations or inequalities. In our exercise, algebraic manipulation is used to work through the inequality. This involves multiplying, dividing, adding, and subtracting terms to isolate the variable, \(x\), on one side of the inequality. For instance, in the positive case, we simplify \[3 < 10 - 4x\] by subtracting 10 from both sides and then dividing by -4, which also requires flipping the inequality sign. This process requires attention to detail, especially when dealing with inequalities where multiplying or dividing by negative numbers reverse the direction of the inequality sign.

Solving Inequalities

Inequalities are solved similarly to equations, but with an added layer of caution. In inequalities, such as \(\frac{3}{|5-2x|}<2\), you must consider the direction of the inequality sign throughout the solution process. Here, solving involves splitting the problem into two cases based on the absolute value condition: \(5-2x > 0\) and \(5-2x < 0\). Each scenario is tackled separately. A key step is maintaining the correct inequality direction when multiplying or dividing by negative numbers, as this reverses the inequality. The ultimate goal is to determine the set of all possible \(x\) values that make the inequality true, which is then expressed in interval notation. This approach ensures a comprehensive understanding and a correct solution for inequalities.