Question

The solution set of \(|3-4 x|>2\) is : (a) \(\left(\frac{7}{4}, \infty\right)\) (b) \(\left(-\frac{1}{4}, \frac{7}{4}\right)\) (c) \(\left(-\infty,-\frac{1}{4}\right) \cup\left(\frac{7}{4}, \infty\right)\) (d) \(\left(-\infty, \frac{1}{4}\right) \cup\left(\frac{5}{4}, \infty\right)\).

Short answer

Question: Solve the inequality \(|3-4x|>2\) and select the correct solution set. a) \(\left(-\infty, -\frac{1}{4}\right) \cup\left(\frac{5}{4}, \infty\right)\) b) \(\left(-\infty, -\frac{1}{4}\right) \cup\left(\frac{1}{4}, \infty\right)\) c) \(\left(-\infty, \frac{1}{4}\right) \cup\left(\frac{1}{4}, \infty\right)\) d) \(\left(-\infty, \frac{1}{4}\right) \cup\left(\frac{5}{4}, \infty\right)\) Answer: (d) \(\left(-\infty, \frac{1}{4}\right) \cup\left(\frac{5}{4}, \infty\right)\)

Step-by-step solution

Break the inequality into two cases

When dealing with the inequality involving the absolute value, we need to consider two cases: when the expression inside the absolute value is positive and when it's negative. So for the given inequality, we will solve the following two inequalities: 1. \(3-4x > 2\) when \(3-4x \ge 0\) 2. \(-(3-4x) > 2\) when \(3-4x < 0\)

Solve the inequality for the first case

Now, we'll solve the inequality \(3-4x > 2\) with the constraint \(3-4x \ge 0\). Subtracting \(2\) from both sides, we get: \(-4x > -1\) Now, divide both sides by \(-4\). Since we are dividing by a negative number, we have to flip the inequality sign: \(x < \frac{1}{4}\) Since the constraint \(3-4x \ge 0\) is true for this case, the solution set for the first case is \(\left(-\infty, \frac{1}{4}\right)\).

Solve the inequality for the second case

Next, we'll solve the inequality \(-(3-4x) > 2\) with the constraint \(3-4x < 0\). Distributing the negative sign, we get: \(-3+4x > 2\) Now, add \(3\) to both sides: \(4x > 5\) Now, divide both sides by 4: \(x > \frac{5}{4}\) Since the constraint \(3-4x < 0\) is true for this case, the solution set for the second case is \(\left(\frac{5}{4}, \infty\right)\).

Combine the solution sets

Now that we have the solution sets for both cases, we need to combine them to obtain the solution set of the original inequality. The solution set is: \(\left(-\infty, \frac{1}{4}\right) \cup\left(\frac{5}{4}, \infty\right)\)

Match with the given options

Comparing the solution set we got to the given options, we can see that our solution set matches with option (d): \(\left(-\infty, \frac{1}{4}\right) \cup\left(\frac{5}{4}, \infty\right)\) So the correct answer is (d).

Key concepts

Inequality Solving

Grasping inequality solving skills is essential for students delving into algebra and beyond. An inequality, unlike an equation, indicates that one side is not necessarily equal to the other, but rather is greater than or less than the other side. In the case of absolute value inequalities like the one in our exercise, the solution involves finding the range of variable values that make the inequality true.

It's critical to remember that inequalities can often have two possible solution sets, as variable expressions can satisfy the inequality in more than one way. This requires splitting the problem into multiple cases, as shown in the detailed problem-solving steps. Each case needs separate consideration because the nature of absolute value functions is such that they measure distance from zero on the number line, which means solutions may occur in two directions—both greater and less than zero. The vital step is to combine the solution sets properly to reflect all possible values of the variable that make the original inequality true.

Absolute Value Equations

Understanding absolute value equations is key to mastering absolute value inequalities. An absolute value refers to the distance of a number from zero on a number line, regardless of direction; it is always non-negative. In the context of the given exercise, the absolute value expression is set to be greater than a positive number. This type of problem is indicative that there is a 'split' in the possible solution sets, where the variable within the absolute value could be on either side of zero on the number line.

The concept requires students to break down the absolute value expression into two separate linear inequalities. This addresses the 'either-or' condition that an absolute value equation suggests: the expression inside the absolute value can be positively or negatively distant from zero, yielding different inequality forms to solve. As demonstrated by the example, careful manipulation of these inequities and understanding the implications of reversing the inequality sign when multiplying or dividing by a negative number, are key to finding the correct solution set.

Quantitative Aptitude

Quantitative aptitude encompasses the ability to handle numerical and logical skill tests, often included in educational assessments or competitive exams. It involves a variety of topics from basic arithmetic to more complex algebraic problems such as the absolute value inequalities discussed in this example. Excelling in quantitative aptitude requires students to have strong problem-solving skills and an intuitive understanding of numbers.

In the context of solving absolute value inequalities, quantitative aptitude is reflected in the ability to logically deduce the steps required to arrive at the solution sets. Moreover, students are often required to apply multiple mathematical concepts simultaneously, such as the order of operations, manipulation of inequalities, and number line analysis. Building a strong quantitative aptitude foundation can significantly help students tackle this wide array of mathematical challenges systematically and efficiently.