Problem 34 - Intermediate Algebra

Question

Solve each inequality. $$ -3 x^{2}-27 \geq 0 $$

Step-by-Step Solution

Answer

There are no real solutions to the inequality.

Step 1: Factor the Inequality

First, we need to factor the quadratic expression on the left side of the inequality. We can factor out \(-3\) from the expression: \(-3(x^2 + 9) \geq 0\).

Step 2: Analyze the Quadratic Expression

The expression \(x^2 + 9\) is always positive because \(x^2\) is non-negative for all real numbers, and adding 9 makes it strictly positive. Thus, \(x^2 + 9 > 0\) for all \(x\).

Step 3: Determine the Sign of the Overall Expression

Since \(x^2 + 9 > 0\), we can analyze the inequality \(-3(x^2 + 9) \geq 0\). The negative sign outside the parentheses will reverse the inequality, making \(-3(x^2 + 9) < 0\). Thus, the expression is always less than zero.

Step 4: Conclude the Solution

The inequality \(-3(x^2 + 9) \geq 0\) has no solution, as \(-3(x^2 + 9)\) is never greater than or equal to zero. Therefore, there are no real values for \(x\) that satisfy the inequality.

Key Concepts

Factorization

Factorization is an essential step in solving quadratic inequalities. It helps simplify the expression, making it easier to work with. To factor a quadratic expression, like in the provided example, you start by identifying common factors.

• In the original inequality, \(-3x^2 - 27 \geq 0\), notice that \(-3\) is a common factor of both terms.
• By factoring \(-3\) out, you can simplify the inequality to \(-3(x^2 + 9) \geq 0\).

Factoring calls for recognizing patterns or using techniques like factoring by grouping or using the quadratic formula when applicable. However, some expressions, like \(x^2 + 9\), remain unfactorable into real linear factors.Understanding which parts of the expression can be factored smoothly helps clarify the inequality making the problem-solving process more straightforward.

Real Numbers

Real numbers are all the numbers you can find on the number line. They include both rational numbers (like integers and fractions) and irrational numbers (like square roots and pi). When dealing with quadratic inequalities, considering real numbers is crucial, as the solutions must fit within this range.

• In the inequality \(x^2 + 9\), since \(x^2\) is positive or zero for any real number, \(x^2 + 9\) is always positive.
• This implies that \(x^2 + 9 > 0\) for all real numbers \(x\), contributing immensely to understanding the subsequent steps of the inequality solution.

By recognizing that quadratic terms become non-negative over real numbers, we can infer key aspects of an inequality without requiring specific number solutions but rather an understanding of number behavior across the real number spectrum.

Inequality Solution Steps

Solving quadratic inequalities involves systematic steps to reach a valid conclusion. Each step is essential and builds on the previous ones. Let's break down the solution steps:

• Step 1: Factoring - Simplify the inequality by factoring. Here, we pulled out \(-3\) from the expression, converting the original inequality to \(-3(x^2 + 9) \geq 0\).
• Step 2: Analyzing the Quadratic Expression - Decipher the behavior of the expression. Since \(x^2 + 9\) is always positive, it influences the inequality by maintaining certain constant positivity.
• Step 3: Determining the Overall Sign - Address how factored terms affect the overall inequality. The negative sign impacts the inequality, changing \(-3(x^2 + 9) \geq 0\) to always being true below zero: \(-3(x^2 + 9) < 0\).
• Step 4: Concluding the Solution - Finally, recognize that no real number solutions exist if the inequality problem remains unsolved under given conditions. In this case, \(-3(x^2 + 9)\) can never meet the equality or surpass zero, hence no solution exists.

By following these systematic steps, we can accurately solve quadratic inequalities by understanding each component's role and how they interact to determine the solution space.