Problem 30 - Precalculus

Question

Solve the equation or inequality. $$-\frac{4}{3}(x-2)^{-\frac{4}{3}}+\frac{8}{9} x(x-2)^{-\frac{7}{3}} \geq 0$$

Step-by-Step Solution

Answer

The inequality has no solution. It never satisfies the non-negativity condition beyond domain restrictions.

Step 1: Identify the Common Factor

Both terms in the inequality have the common factor \((x-2)^{-rac{7}{3}}\). Factor this out from the inequality:\[-(x-2)^{-rac{7}{3}} \left( \frac{4}{3}(x-2)^{1} - \frac{8}{9}x \right) \geq 0\].

Step 2: Simplify the Expression

Simplify the expression inside the parentheses by distributing and combining like terms:\[(x-2) - \frac{2}{3}x \].This simplifies to:\[x - (2x-4)/3 = \frac{3x - 2x + 4}{3} = \frac{x+4}{3}\].

Step 3: Analyze the Factored Inequality

Now the inequality is of the form:\[ -(x-2)^{-rac{7}{3}} \cdot \frac{x+4}{3} \geq 0\].Consider when each factor does not equal zero or when the product has a combined sign change.

Step 4: Solve for Non-zero Conditions

For the expression \(-(x-2)^{-rac{7}{3}} \geq 0\), observe that a negative raised to an algebraic power will maintain its negativity. Since \(-(x-2)^{-rac{7}{3}}\) is negative, it cannot equal or be greater than zero for any real \(x\) in its domain.

Step 5: Verify Conditions and Domain

The term \((x-2)\) raises a potential issue. Since \((x-2)^{-rac{7}{3}}\) involves division by zero when \(x=2\), we exclude \(x = 2\) from the domain. Check only where terms may cancel potential positivity or negativity, if possible.

Step 6: Combine All Conditions

Since \((x-2)^{-rac{7}{3}}\) cannot be positive and the inequality cannot be °C), the expression results in an impossible inequality because both factors never constructively give non-negative values within all possible reals that exclude domain breaches.

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. In algebraic expressions, they play a crucial role in determining how terms are manipulated and simplified. An exponent is written as a small number to the upper right of a base number. For example, in \[x^n\]\(n\) is the exponent, and \(x\) is the base number. Exponents follow specific rules, such as the Power of a Product rule, the Power of a Power rule, and the Negative Exponent rule. Some key rules include:

• Product of Powers: \(a^m \cdot a^n = a^{m+n}\) - Add exponents when multiplying like bases.
• Power of a Power: \((a^m)^n = a^{m \cdot n}\) - Multiply exponents when raising a power to a power.
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\) - A negative exponent indicates reciprocal.

Understanding how exponents transform expressions, like turning subtraction into division, as seen in our inequality where \[(x-2)^{-\frac{7}{3}}\]is involved, is vital for solving and simplifying expressions effectively.

Domain of Functions

The domain of a function includes all the possible input values (or \(x\)-values) for which the function is defined. When determining the domain, consider any restrictions that might cause the function to be undefined.Key aspects to watch out for:

• Division by Zero: Ensure that no variable expression in the denominator equals zero, as the function becomes undefined.
• Even Root of Negative Numbers: For even roots, such as square roots, ensure the radicand (expression inside the root) is non-negative.

In our inequality, the expression contains \[(x-2)^{-\frac{7}{3}}\]which introduces restrictions since \(x-2=0\)makes the expression undefined. As a result, \(x = 2\)is excluded from the domain. Therefore, understanding the domain helps know what values of \(x\) work without causing mathematical errors.

Factoring

Factoring involves breaking down an expression into simpler components or factors. It is a fundamental skill in algebra, especially useful in solving equations and inequalities.There are various methods of factoring, such as:

• Common Factor: Identify and factor out the greatest common factor (GCF) in an expression.
• Difference of Squares: Recognize patterns such as \(a^2 - b^2 = (a-b)(a+b)\).
• Quadratic Trinomials: Factor expressions of the form \(ax^2+bx+c\).

In our inequality problem, factoring was used to simplify the expression by taking out the term \[(x-2)^{-\frac{7}{3}}\]from both parts, turning the complex expression into a product that is easier to handle and analyze. By factoring, you simplify the process of finding solutions or simplifying expressions.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. Understanding how to manipulate these expressions can significantly simplify solving equations and inequalities.Some key points of manipulation include:

• Simplifying: Combine like terms to make the equation neater.
• Distributive Property: \(a(b+c) = ab + ac\) - Used to expand expressions.
• Combining Like Terms: Collect and simplify terms with the same variable part.

In solving inequalities like this one, simplifying and combining like terms in expressions such as \[(x-2) - \frac{2}{3}x\]are crucial. It becomes \(\frac{x+4}{3}\) upon simplification. Recognizing these forms ensures problems are solved efficiently and accurately, making solutions clear and straightforward for all learners.