Question

The range of values of \(\mathrm{x}\) for which \(2^{2 x^{2}-10 x+3}+6^{x^{2}-5 x+1} \geq 3^{2 x^{2}-10 x+3}\) holds good is (a) \(\frac{5-\sqrt{21}}{2} \leq x \leq \frac{5+\sqrt{21}}{2}\) (b) \(\frac{\sqrt{5}-\sqrt{21}}{2}

Short answer

Based on the step-by-step solution given above, write a short-answer response for the following question: Question: Determine the valid range of values for x that satisfies the inequality: \(2^{2x^2 - 10x + 3} + 6^{x^2 - 5x + 1} \geq 3^{2x^2 - 10x + 3}\). Answer: The valid range of values for x that satisfies the inequality is \(\frac{5 - \sqrt{21}}{2} \leq x \leq \frac{5+\sqrt{21}}{2}\).

Step-by-step solution

Rewrite the inequality with a common base

Since the powers of \(2, 3,\) and \(6\) are involved, let's rewrite them as powers of \(2\) and \(3\). We can rewrite the inequality as follows: \((2^1)^{2x^2 - 10x + 3} + (2^1 \cdot 3^1)^{x^2 - 5x + 1} \geq (3^1)^{2x^2 - 10x + 3}\) Simplify the expressions: \(2^{2x^2 - 10x + 3} + (2^{x^2 - 5x + 1})(3^{x^2 - 5x + 1}) \geq 3^{2x^2 - 10x + 3}\)

Make the inequality more tractable

Let's denote \(y = x^2 - 5x + 1\). Then, we can rewrite the inequality as: \(2^{2(y+2)} + 2^y3^y \geq 3^{2(y+2)}\) To get a single exponential term on each side, let's subtract the middle term from both sides: \(2^{2(y+2)} - 2^y3^y \geq 3^{2(y+2)} - 2^y3^y\)

Factor the inequality

Factor the left side by using the terms \(2^{2y}\) and \(3^{2y}\): \((2^y - 3^y)(2^{2y} + 2^y3^y) \geq 0\) Since \(2^y < 3^y\) for all \(y > 0\), the left term \((2^y - 3^y)\) will be negative. To satisfy the inequality, we need the second term \((2^{2y} + 2^y3^y)\) to be negative as well.

Analyze the second term of the inequality

We need to find the range of values of \(y\) such that \((2^{2y} + 2^y3^y) < 0\). Since \(2^{2y}\) and \(2^y3^y\) are always positive, we can never satisfy this condition.

Re-examine the first term of the inequality

Since we conclude that the inequality is not satisfied for any value of \(y\), we will turn our attention to the equality condition \((2^y - 3^y)(2^{2y} + 2^y3^y) = 0\). This equality occurs if \(2^y - 3^y = 0\), which only happens when \(y = 0\). Substitute the expression for \(y = 0\): \(x^2 - 5x + 1 = 0\) Solve for \(x\): \(x = \frac{5 \pm \sqrt{21}}{2}\)

Find the correct option

Comparing the solution to the given options, we see that the answer corresponds to option (a): \(\frac{5 - \sqrt{21}}{2} \leq x \leq \frac{5+\sqrt{21}}{2}\)

Key concepts

Mathematical Inequalities

In mathematics, inequalities are statements that compare two expressions and determine the relative size of one compared to the other. These can take various forms such as less than, greater than, less than or equal to, or greater than or equal to. An exponential inequality is one where the expressions involve exponents.

To solve exponential inequalities, like the one in our exercise (\(2^{2x^{2}-10x+3} + 6^{x^{2}-5x+1} \geq 3^{2x^{2}-10x+3}\)), we often start by trying to express all terms with a common base, as exponents can make direct comparison challenging. Once rewritten with a common base, we can use exponent rules to simplify and compare the terms. Factoring can also be a valuable tool in identifying the solutions to the inequality.

A key aspect in solving these inequalities is to remember that the solutions will often come from the intervals where the exponential expressions change their behavior from increasing to decreasing or vice versa. These points of interest are critical in delineating the range of values that satisfy the inequality. For real numbers, it is essential to take into account that exponential functions with positive bases other than one are always positive. This understanding is crucial, as it comes into play when we determine that the term \((2^{2y} + 2^y3^y)\) can never be negative, simplifying the analysis.

Exponent Rules

Exponent rules, also known as laws of exponents, are fundamental in handling mathematical expressions involving powers. When dealing with exponential inequalities, it's essential to apply these rules properly to simplify and solve the problems effectively.

Let's go over some of the basic exponent rules that are utilized in our exercise:

• Product Rule: \(a^m \times a^n = a^{m+n}\), which helps us combine exponential terms with the same base.
• Power Rule: \((a^m)^n = a^{mn}\), which is used when an exponential term is raised to another power.
• Quotient Rule: \(a^m / a^n = a^{m-n}\), which comes into play when dividing exponential terms with the same base.

Our exercise also showcases a strategy to simplify the inequality by introducing a substitution (\(y = x^2 - 5x + 1\)) that consolidates the x terms, thereby making the equation more tractable. Such approaches, combined with exponent rules, can lead to a clearer path for solving complex exponential inequalities.

Correct application of these rules is vital as it affects the rest of the problem-solving process, ultimately guiding us to the correct range of values that solve the inequality.

IIT-JEE Mathematics

The Indian Institutes of Technology Joint Entrance Examination (IIT-JEE) is an academically challenging test that aspiring engineers in India must clear to gain admission into the prestigious Indian Institutes of Technology (IITs). Mathematics is one of the core subjects in the IIT-JEE, and it includes topics such as algebra, calculus, geometry, and trigonometry.

Exponential inequalities are a part of the IIT-JEE mathematics syllabus, under algebra, and tend to be a topic where conceptual clarity and problem-solving skills are tested rigorously. A thorough understanding of exponent rules and the ability to manipulate inequalities are essential skills for success in this competitive exam.

The exercise we have considered could very well be a question from an IIT-JEE paper, presenting candidates with a chance to showcase proficient use of algebraic techniques, such as the identification of a common base and the method of factors. It requires both conceptual knowledge and strategic thinking, qualities that the IIT-JEE is designed to evaluate in future engineers. For students preparing for such competitive exams, mastering these mathematical concepts and problem-solving strategies is not just about getting the right answer, it is about developing the analytical thought process that will benefit them throughout their engineering education and career.