Problem 90 - Algebra and Trigonometry

Question

What's Wrong Here? It is tempting to try to solve an inequality like an equation. For instance, we might try to solve \(1<3 / x\) by multiplying both sides by \(x,\) to get \(x<3\) so the solution would be \((-\infty, 3) .\) But that's wrong; for example, \(x=-1\) lies in this interval but does not satisfy the original inequality. Explain why this method doesn't work (think about the sign of \(x\) ). Then solve the inequality correctly.

Step-by-Step Solution

Answer

Multiplying by \(x\) without considering its sign is incorrect; valid solution interval is \((0,3)\).

Step 1: Understanding the Issue

The step where the mistake occurs is when both sides of the inequality \(1 < \frac{3}{x}\) are multiplied by \(x\). This operation is problematic because the sign of \(x\) affects the direction of the inequality.

Step 2: Properties of Inequality

When multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed. In the faulty solution, multiplying by \(x\) does not account for whether \(x\) is positive or negative, which affects the inequality differently for \(x > 0\) and \(x < 0\).

Step 3: Analyzing Sign of x

Assess the behavior of the inequality based on whether \(x\) is positive or negative. For \(x > 0\), multiplying by \(x\) keeps the inequality direction unchanged, while for \(x < 0\), it would need to be reversed.

Step 4: Testing Different Cases

Consider \(x > 0\): for \(1 < \frac{3}{x}\), multiply by \(x\) to get \(x < 3\). For \(x < 0\), \(1 < \frac{3}{x}\) becomes \(x > 3\), but since \(x\) is negative, it does not satisfy the inequality.

Step 5: Check Zero and Undefined Points

Notice that at \(x = 0\), \(\frac{3}{x}\) is undefined, so zero is not in the solution set. \(x\) cannot be zero.

Step 6: Determine Correct Interval

From previous considerations, \(x > 0\) and \(x < 3\) form the proper solution, since the inequality \(1 < \frac{3}{x}\) holds true when \(0 < x < 3\).

Step 7: Conclusion

The correct solution to the inequality \(1 < \frac{3}{x}\) is \(x \in (0, 3)\), as for these values of \(x\), \(\frac{3}{x}\) remains greater than 1.

Key Concepts

Inequality Properties

When dealing with inequalities, it's crucial to understand how they differ from equations. Inequalities allow us to compare two values, showing if one is greater than, less than, or simply not equal to the other.
These relationships, unlike equations, can be affected by the operations we perform.

• When you add or subtract the same number on both sides of an inequality, the relationship between the quantities doesn't change.
• However, multiplying or dividing both sides by a positive number leaves the direction of the inequality unchanged.
• On the contrary, if you multiply or divide by a negative number, the direction of the inequality must be flipped for it to still be true.

This crucial property is often a stumbling block, as seen in the exercise where multiplying by an unknown sign leads to errors.

Sign of Variable in Inequalities

The sign of a variable in inequalities is pivotal in deciding how an inequality transforms. Imagine dealing with an inequality like the one given: if we multiply by a variable like \(x\), whose sign we do not know, we risk flipping the inequality direction incorrectly.
This is because:

• If \(x > 0\), multiplying by \(x\) does not change the inequality direction.
• If \(x < 0\), the entire inequality flips direction, meaning \(a < b\) becomes \(ax > bx\).

In our original exercise, not recognizing whether \(x\) is positive or negative led to a false conclusion. To avoid this, it helps to first determine the sign of \(x\) or solve the inequality within known sign intervals.

Solving Rational Inequalities

Solving rational inequalities involves understanding both the algebraic manipulation of inequalities and the peculiarities of rational expressions. Here’s a simple process:

• Identify critical points where the expression is undefined or equals zero. For a fraction like \(\frac{3}{x}\), zeroes of \(x\) create undefined points.
• Consider different intervals formed by these critical points. For instance, intervals for \(x\) in our problem could be \((-\infty, 0)\), \((0, 3)\), and \((3, \infty)\).
• Test each interval to determine where the inequality holds true. Use test points, excluding undefined or boundary points that do not meet the inequality's requirements.

In our exercise, the correct approach revealed that the solution for \(1 < \frac{3}{x}\) is within the interval \(x \in (0, 3)\). This is because the inequality remains valid only when \(\frac{3}{x}\) exceeds 1 in this interval.