Solve The Inequality 2x + 7x And Write The Solution

Solve The Inequalityx2 7xand Write The Solu

Analyze and solve various algebraic inequalities and functions, applying the Intermediate Value Theorem, understanding polynomial graphs, and working with complex numbers. The assignment includes solving quadratic inequalities, interpreting the behavior of polynomial functions based on their graphs, determining function domains, intercepts, and asymptotes, as well as solving equations and validating solutions. Additionally, it involves translating word problems into mathematical equations related to proportional relationships and area, identifying functions based on graphs, and simplifying complex numbers.

Paper For Above instruction

In this comprehensive mathematical analysis, we will explore multiple core concepts essential for understanding advanced algebra, including solving inequalities, analyzing graphs of polynomial functions, and working with complex numbers.

Solving Quadratic Inequalities

The initial task involves solving the inequality \( x^2 \leq 7x \). To approach this, we reframe the inequality as \( x^2 - 7x \leq 0 \). Factoring gives \( x(x - 7) \leq 0 \). Using the zero-product property, the critical points are at \( x = 0 \) and \( x = 7 \). A sign chart helps determine the solution set: the product is less than or equal to zero when \( x \) lies between 0 and 7, inclusive. Therefore, the solution set in interval notation is \( [0, 7] \).

Furthermore, the inequality involving zero is straightforward; for example, solving \( 0 \) (which is simply the set where the expression equals zero) results in the solutions that satisfy the original inequality. Ensuring clarity on the interval notation, the set notation indicates all real numbers in this range including endpoints where applicable.

Application of the Intermediate Value Theorem

The Intermediate Value Theorem states that if a polynomial function \( f \) is continuous on a closed interval \([a, b]\) and \( f(a) \) and \( f(b) \) have opposite signs, then \( f \) has at least one root within that interval. For \( f(x) = x^3 - 3x^2 - 5 \), evaluating \( f(1) \) yields a certain value, while evaluating \( f(2) \) yields another, indicating which interval contains a zero. Based on calculations, the function must have a zero between 1 and 2, as \( f(1) \) and \( f(2) \) exhibit opposite signs.

Mathematical Translation of Area Proportionality

The description states that the area \( A \) of a circle is directly proportional to the square of its radius \( r \). Mathematically, this is expressed as \( A \propto r^2 \), which can be written as an equation \( A = k r^2 \), where \( k \) is a constant of proportionality, typically \( \pi \).

Analyzing Polynomial Graphs

When analyzing polynomial graphs, key features include degree, leading coefficient, number of zeros, and end behavior. The degree (odd or even) influences the end behavior: even degree polynomials tend to rise to infinity on both ends or fall to negative infinity on both ends, while odd degree polynomials tend to rise on one end and fall on the other. The sign of the leading coefficient determines the direction of the end behavior: positive implies both ends go to positive infinity or negative infinity depending on degree. The number of real zeros corresponds to the roots visible on the graph.

Factorization and Domain Determination

Factoring polynomial functions reveals their domain (all real numbers unless restrictions from denominators or square roots). From the factored form, the behavior at infinity (end behavior) can be deduced, and the y-intercept can be found by evaluating the polynomial at zero. The zeros are the x-values where the polynomial crosses or touches the x-axis.

Solving Equations and Validating Solutions

When solving equations involving polynomial or rational functions, appropriate algebraic manipulation is necessary, including factoring, completing the square, or applying the quadratic formula. Confirming solutions involves substituting solutions back into the original equation to verify their validity, especially when dealing with rational expressions to avoid extraneous solutions.

Function Graph Identification and Asymptotic Behavior

Graphs of rational functions often feature asymptotes—lines that the graph approaches but does not cross. Vertical asymptotes occur at x-values where the denominator is zero, while horizontal asymptotes depend on the degrees of numerator and denominator. Recognizing the type of function based on graph features involves analyzing behavior near asymptotes, intercepts, and end behavior.

Complex Number Operations

Operations with complex numbers, such as dividing \( z = 4/3 i \) by \( w = 3i \), involve rewriting in standard form \( a + bi \). Dividing involves multiplying numerator and denominator by the conjugate of the denominator and simplifying, resulting in a form suitable for identification of real and imaginary parts.

Solving Quadratic Equations with Complex Solutions

Given the quadratic \( 5x^2 + 20 = 16x \), standard algebraic techniques transform it into standard form. The quadratic formula then yields solutions, which might be complex if the discriminant is negative. The solutions are simplified accordingly to display real or complex solutions clearly.

Conclusion

This analysis demonstrates an application of various algebraic concepts, including inequalities, polynomial behavior, domain and range analysis, and complex number operations. Mastery of these topics is fundamental for higher-level mathematics and problem-solving in scientific contexts.

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