Question

Which of the following inequalities is the solution to the inequality \(2 x^2+1x>0\) B. \(-1

Short answer

A. \(\0 < x < 1/2\)

Step-by-step solution

- Simplify the Inequality

Start by simplifying the given inequality: \[2x^2 + 1 < x + 1\]Subtract 1 from both sides:\[2x^2 < x\]

- Rearrange the Inequality

Rearrange the inequality so that one side is 0:\[2x^2 - x < 0\]

- Factor the Quadratic Expression

Factor the quadratic expression:\[x(2x - 1) < 0\]

- Find the Critical Points

Set each factor equal to zero to find the critical points:\[x = 0\] and \[2x - 1 = 0\]Solve for x in the second equation:\[2x - 1 = 0\]\[x = \frac{1}{2}\]

- Determine the Intervals

Determine the intervals around the critical points:\[-\text{to}\frac{1}{2}, 0\text{to}\frac{1}{2}\]

- Test the Intervals

Test points from each interval in the inequality to determine where it is true:Test \(x = -1\) in \(x(2x - 1) < 0\): \[-1(2(-1) - 1) = -1(-3) = 3 > 0\] (Not valid)Test \(x = \frac{1}{4}\) in \(x(2x - 1) < 0\): \[\frac{1}{4}\bigg(2\bigg(\frac{1}{4}\bigg) - 1\bigg) = \frac{1}{4}(\frac{1}{2} - 1) = \frac{1}{4}(-\frac{1}{2}) = -\frac{1}{8} < 0\] (Valid)Test \(x = 1\) in \(x(2x - 1) < 0\): \[1(2(1) - 1) = 1(1) = 1 > 0\] (Not valid)

- Conclude the Solution

Between 0 and \(\frac{1}{2}\), the inequality holds true. Thus, the solution is:\[0 < x < \frac{1}{2}\]

- Choose the Correct Option

Compare the solution with the given options:Option A is the correct answer: \[0 < x < \frac{1}{2}\]

Key concepts

GMAT preparation

Preparing for the GMAT involves mastering various mathematical concepts, including inequalities. Inequalities on the GMAT can range from simple linear inequalities to more complex quadratic inequalities. One effective preparation strategy is to practice solving different types of inequalities. Understanding the steps and concepts behind these problems is crucial.
Key aspects for GMAT preparation include:

• Familiarizing yourself with different inequality types and solving methods.
• Regular practice to increase problem-solving speed and accuracy.
• Focus on step-by-step techniques for isolating and testing variables.

Being methodical and thorough in your approach will help you achieve a high score. By practicing these exercises, you'll build the confidence needed to tackle similar problems effectively on the test day.

inequality solving steps

Solving inequalities involves several steps to rearrange and simplify the expression. Here are the detailed steps illustrated by our example problem:

• Simplify the given inequality. For our example, we started with \(2x^2 + 1 < x + 1\) and simplified it to \(2x^2 < x\).
• Rearrange the inequality to have zero on one side. This usually means moving all terms to one side. In our case, we got \(2x^2 - x < 0\).
• Factor the expression if possible. Our expression \(2x^2 - x\) factors to \(x(2x - 1) < 0\).
• Identify critical points by setting each factor to zero. We found critical points \(x = 0\) and \(x = \frac{1}{2}\).
• Test the intervals around the critical points to see where the inequality holds true. For intervals \( (-\text{to}0)\text{to}\frac{1}{2} \), we tested several points.
• Conclude and interpret the solution based on valid intervals. Our final solution was \(0 < x < \frac{1}{2}\).

Practicing these steps will help you solve similar inequalities with confidence.

quadratic inequalities

Quadratic inequalities involve polynomial expressions where the highest degree is two. Solving them typically involves finding where the quadratic expression is less than or greater than zero. Here’s how you can approach these problems:

• Start by simplifying the inequality and moving all terms to one side, so you can have an expression like \(ax^2 + bx + c < 0\) or \(ax^2 + bx + c > 0\).
• Next, factor the quadratic expression if possible. Factoring helps decompose the quadratic into linear components which are easier to evaluate.
• Identify the critical points, which are the roots of the equation. These are found by setting each factor to zero.
• Determine the intervals around these critical points. The quadratic expression will change its sign only at these points.
• Test points from each interval in the original inequality to determine where it holds true. This helps in finding the range where the inequality is valid.
• Finally, state your solution based on the intervals that satisfied the original inequality.

Understanding these concepts is crucial for solving quadratic inequalities efficiently on the GMAT. Always double-check your critical points and tested intervals to avoid mistakes and ensure you have the correct solution.