Math Problem Statement

Solve the inequality in terms of intervals. (Enter your answer using interval notation.)

1\x < 2

Solution

To solve the inequality 1x<2\frac{1}{x} < 2, follow these steps:

Step 1: Multiply both sides of the inequality by xx

We need to handle the inequality by isolating xx. However, be cautious when multiplying or dividing by xx, as it affects the direction of the inequality depending on whether xx is positive or negative.

So, we will break the solution into two cases:

Case 1: x>0x > 0

When x>0x > 0, multiplying by xx does not change the inequality direction. So, the inequality becomes:

1<2x1 < 2x

Now, solve for xx:

12<x\frac{1}{2} < x

This simplifies to:

x>12x > \frac{1}{2}

Case 2: x<0x < 0

When x<0x < 0, multiplying by xx flips the inequality direction. So, the inequality becomes:

1>2x1 > 2x

Solving for xx:

12>x\frac{1}{2} > x

This simplifies to:

x<12x < \frac{1}{2}

Step 2: Combine the results

We now combine the results of both cases:

  • For x>0x > 0, we have x>12x > \frac{1}{2}.
  • For x<0x < 0, we have x<12x < \frac{1}{2}.

Thus, the solution in interval notation is:

(,0)(12,)(-\infty, 0) \cup \left(\frac{1}{2}, \infty\right)

Final Answer:

The solution in interval notation is: (,0)(12,)(-\infty, 0) \cup \left(\frac{1}{2}, \infty\right)

Would you like further details or have any specific questions on the solution?

Related Questions:

  1. How would the solution change if the inequality was 1x2\frac{1}{x} \geq 2?
  2. What happens if the inequality is 1x>2\frac{1}{x} > 2?
  3. Can we solve this inequality without considering different cases for xx?
  4. How would the solution change if we had 1x2\frac{1}{x} \leq 2?
  5. What are the restrictions on xx for the expression 1x\frac{1}{x} to be defined?

Tip:

Always be cautious when multiplying or dividing by a variable in inequalities, as it may flip the inequality depending on the sign of the variable.

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Math Problem Analysis

Mathematical Concepts

Inequalities
Algebra
Interval Notation

Formulas

1/x < 2
Inequality multiplication rule for x > 0 and x < 0

Theorems

Multiplying or dividing by a variable in inequalities requires case analysis based on the sign of the variable.

Suitable Grade Level

Grades 9-11