Problem 58 - Algebra for College Students

Question

Solve each inequality. State the solution set using interval notation when possible. \(x^{2}+1 \leq 0\)

Step-by-Step Solution

Answer

The solution set is empty: \(\emptyset\).

Step 1: Understand the Inequality

We are given the inequality: \(x^{2} + 1 \leq 0\)This inequality implies that we need to find the values of \(x\) for which the expression \(x^{2} + 1\) is less than or equal to zero.

Step 2: Analyze the Expression

\(x^{2} + 1\) is a quadratic expression. Evaluate its components: \(x^{2}\) denotes the square of \(x\), which is always non-negative (\(x^{2} \geq 0\)), and adding 1 makes it positive (\(x^{2} + 1 \geq 1\)).

Step 3: Determine Feasibility

Since \(x^{2} \geq 0\), the minimum value of \(x^{2} \) is 0. Therefore, \(x^{2} + 1\) must always be \(\geq 1\). Consequently, there is no value of \( x \) that will make \(x^{2} + 1 \leq 0\).

Step 4: State the Solution in Interval Notation

Since there are no solutions that satisfy the inequality, the solution set in interval notation is the empty set, denoted by \(\emptyset\).

Key Concepts

inequality solution

An inequality solution involves determining a range or set of values that satisfy the given inequality. In this exercise, you are asked to solve the inequality:
\(x^2 + 1 \leq 0\). We want to find values of x that make the expression \(x^2 + 1\) less than or equal to zero. To solve it, follow these steps:

• First, understand the expression and what it represents.
• Second, analyze its components to see if there are values that make the inequality true.

Specifically, for \(x^2\), because squaring any real number always results in a non-negative value, adding 1 to this non-negative value will always result in a number greater than or equal to 1. Since this result can never be less than or equal to 0, there are no values of x that can satisfy the inequality. Therefore, the solution is the empty set, written as \emptyset.

interval notation

Interval notation is a way of writing subsets of the real numbers. This notation uses parentheses and brackets to represent intervals. For example:

• (a, b) represents all numbers between a and b, but not including a and b.
• [a, b] includes the endpoints a and b.
• (a, b] includes b but not a.
• [a, b) includes a but not b.

When the solution to an inequality is a range of values, interval notation provides a clear and concise way to express these values. For example, if the solution was all x greater than 2 and less than 5, we would write it as (2, 5). In our specific exercise, because no real number satisfies the inequality \(x^2 + 1 \leq 0\), the solution is the empty set, which is written as \emptyset.

quadratic expressions

Quadratic expressions are algebraic expressions of the form \(ax^2 + bx + c\), where a, b, and c are constants. The variable x here is raised to the second power, making the expression quadratic. Some characteristics of quadratic expressions include:

• The graph of a quadratic expression is a parabola.
• Depending on the sign of the constant a, the parabola can open upwards (a > 0) or downwards (a < 0).
• Quadratic expressions can have zero, one, or two real roots depending on the discriminant (\(b^2 - 4ac\)).

In our exercise, the given expression is \(x^2 + 1\). Here, it is clear that:

• a = 1, b = 0, and c = 1.
• The discriminant (b^2 - 4ac) is 0^2 - 4*1*1 = -4, a negative number, indicating that there are no real roots.
• The quadratic expression \(x^2 + 1\) is always greater than or equal to 1 since the minimum value of \(x^2\) is 0.

This information helps us conclude that the inequality \(x^2 + 1 \leq 0\) has no solution in the real numbers.