Question

For which values of \(x\) is \(5 x-x^2-3\) greater than \(x ?\) (A) \(1>x>3\) (B) \(0>x>2\) (C) \(-3

Short answer

The correct interval is 1 < x < 3.

Step-by-step solution

- Set up the inequality

Start by setting up the given inequality: i.e.5x - x^2 - 3 > x

- Simplify the inequality

Subtract x from both sides to simplify:5x - x^2 - 3 - x > 0Combine like terms:- x^2 + 4x - 3 > 0

- Form a quadratic equation

Consider the corresponding equality to find roots of the quadratic equation:- x^2 + 4x - 3 = 0

- Solve the quadratic equation

Use the quadratic formula to solve for x:i.e.\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where a = -1, b = 4, and c = -3.Solving this, you will get:x = 1 and x = 3

- Identify the inequality intervals

The quadratic equation roots divide the number line into three intervals:1. x < 12. 1 < x < 33. x > 3Test for values inside each of these intervals to see where the inequality holds.

- Test the interval where the inequality holds

Choose values within these intervals and test them in the inequality1. For x = 0 (x < 1): - 0^2 + 4(0) - 3 = -3 (not greater than 0)2. For x = 2 (1 < x < 3):- (2)^2 + 4(2) - 3 = 5 (greater than 0)3. For x = 4 (x > 3):- (4)^2 + 4(4) - 3 = -3 (not greater than 0)Hence, the inequality is satisfied within the interval x lies between 1 and 3

Key concepts

Inequalities

When solving GMAT inequality problems, it's crucial to understand how inequalities work. Inequalities show the relationship between two expressions. For instance, in the provided exercise, the inequality is given as: \( 5x - x^2 - 3 > x \).

• This inequality indicates that one side must be greater than the other.
• The first step is always to transform the given inequality into a more manageable form.

Inequalities help us understand how values on both sides relate to each other. We also need to make sure that the inequality holds when we solve it.

Quadratic Equations

Quadratic equations are equations that involve terms with the variable raised to the second power, i.e., \(ax^2 + bx + c = 0\). In our problem, after simplifying, we get the quadratic equation: - \(x^2 + 4x - 3\). The two key aspects of quadratic equations are:

• Roots: The values of \(x\) that satisfy the equation. From our example, solving for x in \(- x^2 + 4x - 3 = 0\) gives us the roots \(x = 1\) and \(x = 3\).
• Graph: The parabola represented by the quadratic equation. The roots divide our equation into intervals which we need to test for where the inequality holds.

Problem-Solving Steps

Following structured problem-solving steps is key for tackling GMAT problems effectively:

• Set up the inequality: Begin by ensuring all terms are on one side of the inequality.
• Simplify: Combine like terms to simplify the expression. In the exercise, subtract x to transform \(5x - x^2 - 3 > x\) into \(- x^2 + 4x - 3 > 0 \).
• Form a quadratic equation: Equate the inequality to zero (\(- x^2 + 4x - 3 = 0\)) to solve for roots using the quadratic formula.

Finally, identify the intervals divided by these roots, and test values from each interval to find where the inequality holds. This method improves accuracy and understanding of inequality solutions.

Test Prep

Effective test preparation for GMAT includes mastering concepts like inequalities and quadratic equations. Here are helpful tips:

• Practice Regularly: Solve various inequality problems to become comfortable with different types.
• Review Key Formulas: Memorize the quadratic formula and the steps to solve inequalities.
• Analyze Mistakes: Understand where you went wrong in practice problems to avoid repeating errors.

Remember, the GMAT not only tests your mathematical skills but also your problem-solving approach and logical thinking. Efficiently practicing, structuring your approach, and reflecting on mistakes will lead to better performance.