Question

Is \(x\) positive or negative? (1) \(x^3+8=0\) (2) \(x^2-4=0\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

A. 1 alone, not 2 alone

Step-by-step solution

Analyzing Statement 1

Given is the equation \(x^3 + 8 = 0\). Subtract 8 from both sides to get: \(x^3 = -8\). Now, take the cube root of both sides: \(x = \sqrt[3]{-8}\). Since the cube root of -8 is -2, \(x = -2\). Therefore, \(x\) is negative.

Analyzing Statement 2

Given is the equation \(x^2 - 4 = 0\). Add 4 to both sides to get: \(x^2 = 4\). Now, take the square root of both sides: \(x = \sqrt{4}\) or \(x = -\sqrt{4}\). This gives us \(x = 2\) or \(x = -2\). Therefore, \(x\) could be positive or negative.

Determine Sufficiency

From Statement 1, we determined that \(x\) is definitely negative. From Statement 2, we determined that \(x\) could be either positive or negative. Therefore, only Statement 1 is sufficient to determine whether \(x\) is positive or negative.

Key concepts

Inequalities in Algebra

Inequalities in algebra are a crucial concept for understanding mathematical relationships. They tell us how one side of an expression compares to the other. Unlike equations, which state that two expressions are equal, inequalities show that one expression is greater than or less than the other.
For instance, consider the inequality: 3x + 2 > 11. To solve it, you need to isolate x by performing arithmetic operations:

• Subtract 2 from both sides: 3x > 9
• Divide both sides by 3: x > 3

This means that the variable x must be greater than 3 for the inequality to hold true. Understanding how to manipulate inequalities helps in solving more complex algebraic problems.

Equation Solving

Solving equations is about finding the value(s) of the variable that satisfy the given equation. You often need to isolate the variable to do this effectively. Let's revisit the exercise, where we deal with specific equations.
Take the equation from Statement 1 of the exercise: \r \(x^3 + 8 = 0\).

• First, subtract 8 from both sides: \(x^3 = -8\).
• Next, take the cube root of both sides: \(x = \sqrt[3]{-8}\).

This gives us x = -2. Here, x is clearly isolated and solved.
In Statement 2, the equation is: \(x^2 - 4 = 0\).

• Add 4 to both sides: \(x^2 = 4\).
• Take the square root of both sides: \(x = \sqrt{4}\) or \(x = -\sqrt{4}\).

This gives us two possible values: x = 2 or x = -2. Therefore, x could be positive or negative, showing different characteristics between cube and square roots.

Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In mathematical terms, the cube root of a number x is denoted as \sqrt[3]{x}.
Consider the cube root operation we performed in Statement 1: We had \(x^3 = -8\).
The cube root of -8 is found by determining what number, when cubed, gives -8. The answer is -2 because \(-2 * -2 * -2 = -8\). Thus, \(\sqrt[3]{-8} = -2\).
When working with cube roots, it's important to remember that the cube root of a negative number is also negative. This is different from square roots, where the root of a positive number has both positive and negative values.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. It's denoted as \sqrt{x}. In algebra, square roots can sometimes produce two results: one positive and one negative.
For example, let's revisit Statement 2 where we had the equation \(x^2 = 4\).
Taking the square root of both sides, we get:

• \(x = \sqrt{4}\) or \(x = -\sqrt{4}\).

This (\(\sqrt{4}\) and -\(\sqrt{4}\)) means x could be either 2 or -2. This is because both \(2 * 2 = 4\) and \(-2 * -2 = 4\).
Unlike cube roots, square roots of positive numbers always yield two roots: one positive and one negative. Understanding this difference is key to solving equations involving square roots effortlessly.