Problem 42 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

No real solution exists as \(x^2\) cannot be less than -9.

Step 1: Rearrange the Inequality

Start by rearranging the given inequality:\[-3x^2 - 27 \geq 0\]Add 27 to both sides to isolate the term involving \(x\):\[-3x^2 \geq 27\]

Step 2: Remove Negative Coefficients

Divide both sides of the inequality by -3, remembering that dividing by a negative number reverses the inequality:\[x^2 \leq -9\]

Step 3: Interpret the Inequality

The inequality \(x^2 \leq -9\) suggests that the square of a real number \(x\) is less than or equal to a negative number. Recall that squares of real numbers are always non-negative, therefore \(x^2 \leq -9\) has no real solutions.

Key Concepts

Real Numbers

Real numbers are an essential concept in understanding mathematics. They include all the numbers that can be found on the number line. These numbers include whole numbers (like 0, 1, and 3), fractions (like \( \frac{1}{2} \)), and decimals (like 3.14). Real numbers also include both positive numbers and negative numbers, as well as zero.
It is important to note that real numbers do not include imaginary numbers. Real numbers can be categorized into several different types:

• Natural Numbers: These are the counting numbers like 1, 2, 3, and so on.
• Whole Numbers: These include all natural numbers and zero.
• Integers: These are all whole numbers, including negative numbers, zero, and positive numbers.
• Rational Numbers: These numbers can be expressed as the quotient or fraction of two integers.
• Irrational Numbers: Numbers like \( \pi \) and \( \sqrt{2} \) that cannot be expressed as simple fractions.

Understanding real numbers is crucial for solving inequalities and quadratic equations, as they provide the set of numbers within which we typically seek solutions.

Inequality Solution

An inequality essentially tells us how two values relate to one another. Unlike an equation which asserts that two expressions are equal, an inequality indicates that one is greater or smaller. Solving inequalities involves finding the range of values that satisfy the inequality condition.
To solve inequalities, similar to solving equations, we perform operations such as addition, subtraction, multiplication, and division. However, a critical point to remember is when you multiply or divide by a negative number, the inequality sign must be reversed. This means if you’re solving an inequality like \[-3x \geq 9\], when you divide both sides by \(-3\), the inequality becomes\[x \leq -3\].
There are different types of inequalities:

• Strict Inequality: Like \(x < y\) or \(x > y\), indicates that \(x\) is strictly less than or greater than \(y\).
• Non-strict Inequality: Like \(x \leq y\) or \(x \geq y\), includes the possibility that \(x\) is equal to \(y\).

Using these rules effectively helps in finding a solution to inequalities, especially when dealing with quadratic inequalities.

Quadratic Equations

Quadratic equations are of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. These equations are called quadratic because the highest exponent of the variable \(x\) is 2, which defines the degree of the equation.
One of the primary tools for solving quadratic equations is factoring. Additionally, the quadratic formula can also be used, which is given as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The solutions obtained through this process for the variable \(x\) are the roots or zeros of the equation.
Understanding quadratic equations is vital when dealing with quadratic inequalities. Here's how they relate:

• Quadratic inequalities deal with expressions that have a square term and involve inequalities (\(<, >, \leq, \geq\)) instead of equal signs.
• The solutions describe the intervals on the real number line where the inequality holds true.

In the exercise you worked on, solving \(-3x^2 - 27 \geq 0\) involved recognizing that when you try to solve \(x^2 \leq -9\), there are no real solutions, since no real number squared gives a negative result.