Question

If \(m x^{2}

Short answer

(1) \(x \in\left(\frac{n}{m}, \infty\right)\) (2) \(x \in\left(0, \frac{n}{m}\right)\) (3) \(x \in\left(-\infty, \frac{n}{m}\right)\) Answer: (1) \(x \in\left(\frac{n}{m}, \infty\right)\)

Step-by-step solution

(Step 1: Rewrite the inequality in terms of x)

First, let's rewrite the inequality as follows: \(x^2 < \frac{n}{m}x\)

(Step 2: Case m > 0 and n < 0)

If \(m > 0\) and \(n < 0\), let's analyze the inequality in this case: \(x^2 < \frac{n}{m}x\) Now, let's divide both sides of the inequality by \(x\) assuming \(x\) is positive: \(x < \frac{n}{m}\) Since \(n\) is negative, we get that \(x\) is positive and less than a negative number, which is not possible. Therefore, no interval in this case is valid.

(Step 3: Case m < 0 and n > 0)

If \(m < 0\) and \(n > 0\), let's analyze the inequality in this case: \(x^2 < \frac{n}{m}x\) Same as before, let's divide both sides of the inequality by \(x\) assuming \(x\) is positive: \(x > \frac{n}{m}\) Since \(m\) is negative and \(n\) is positive, the ratio \(\frac{n}{m}\) is negative, and therefore, \(x\) is a positive number greater than a negative number, meaning \(x \in \left(\frac{n}{m}, \infty\right)\). So, this inequality is valid for option (1).

(Step 4: Conclusion)

From the analysis of the two possible cases for \(m\) and \(n\), we found that the inequality \(mx^2 < nx\) is valid for \(x \in \left(\frac{n}{m}, \infty\right)\) when \(m\) and \(n\) have opposite signs. Therefore, the answer is (1) \(\mathrm{x} \in\left(\frac{\mathrm{n}}{\mathrm{m}}, \infty\right)\).

Key concepts

Quadratic Inequalities

Quadratic inequalities are mathematical expressions involving a quadratic expression set less than or greater than another expression, typically zero. They are of the form ax^2 + bx + c > 0 or ax^2 + bx + c < 0 where a, b, and c represent coefficients.

Solving a quadratic inequality requires finding the range of values for the variable x that makes the inequality true. This often involves factoring the quadratic expression, identifying the roots or zeros of the corresponding quadratic equation, and performing a sign analysis to determine where the expression is positive or negative. The solution is typically expressed as an interval or a union of intervals, representing the set of all possible x values that satisfy the inequality.

Mathematical Intervals

Mathematical intervals are used to describe a set of numbers that lie between two endpoints. Intervals can be closed, open, or a combination of both. An open interval, such as (a, b), includes all the numbers between a and b but not a or b themselves. A closed interval, such as [a, b], includes all numbers between a and b and also includes the endpoints a and b.

Intervals are crucial in communicating the solutions to inequalities, as they succinctly convey the range of permissible values. For example, if a solution to an inequality is x > 3, we can express this using the interval notation (3, ∞).

Signs of Coefficients

The signs of coefficients in a quadratic expression, such as the a, b, and c in ax^2 + bx + c, critically influence the shape and position of the parabola when the expression is graphed. The sign of the leading coefficient a, determines whether the parabola opens upwards (positive a) or downwards (negative a).

In the context of inequalities, the signs of coefficients also determine the solution intervals. When analyzing inequalities, a change in sign can flip the direction of the inequality, affecting which values of x satisfy the condition. Recognizing the importance of the signs helps simplify and solve inequalities effectively.

Solving Inequalities

Solving inequalities involves finding all the values of the variable that make the inequality true. The steps typically include isolating the variable on one side of the inequality, simplifying the expression, and considering the critical points where the expression equals zero.

It's also essential to consider special cases, such as multiplication or division by a negative number, which reverses the inequality sign. To solve quadratic inequalities, factorization, completing the square, or employing the quadratic formula may be necessary before analyzing the sign of the expression over different intervals. Finally, solutions to inequalities are often expressed using interval notation, which concisely shows the range of values that satisfy the inequality.