Problem 57 - Algebra for College Students

Question

Solve each inequality. State the solution set using interval notation when possible. \(x^{2}+4 \geq 0\)

Step-by-Step Solution

Answer

The solution set is \((-\infty, \infty)\).

Step 1: Analyze the inequality

The given inequality is \(x^2 + 4 \geq 0\). We need to determine for which values of \(x\) the inequality holds true.

Step 2: Identify the nature of the quadratic expression

Consider the quadratic expression inside the inequality: \(x^2 + 4\). Notice that \(x^2\) is always non-negative (i.e., \(x^2 \geq 0\) for all real numbers \(x\)). Also, adding 4, a positive constant, will always result in a positive value. Therefore, \(x^2 + 4 \) is always greater than or equal to zero for all real numbers \(x\).

Step 3: Conclude the solution set

Since \(x^2 + 4\) is always greater than or equal to zero for all \(x\), the inequality \(x^2 + 4 \geq 0\) is true for all real numbers.

Step 4: Express the solution in interval notation

The solution set for the inequality covers all real numbers. Thus, in interval notation, the solution is \((-\infty, \infty)\).

Key Concepts

Headline of the respective core concept

Quadratic inequalities involve expressions where a quadratic polynomial is compared to a value using inequality symbols such as <, >, ≤, or ≥. Identifying whether the quadratic expression is always positive, always negative, or changes sign depending on the variable is key. In our example, we need to find when the expression \(x^2 + 4 \geq 0\). Start by analyzing the quadratic part: \(x^2\). This part is always non-negative because squaring any real number (positive, negative, or zero) gives a non-negative result. Adding 4 to \(x^2\) means the expression will always be at least 4, a positive value.

Headline of the respective core concept

The solution set of an inequality represents all the possible values of the variable that make the inequality true. For \(x^2 + 4 \geq 0\), we determined that \(x^2 + 4\) is always non-negative for all real values of \(x\). This implies that every real number will satisfy the inequality. Therefore, the solution set includes all real values of \(x\). If asked for which values make the inequality true, the answer would be 'all real numbers.'

Headline of the respective core concept

Interval notation is a way of writing subsets of the real number line. It shows the start and end of a range and uses parentheses or brackets to indicate whether endpoints are excluded or included. The solution set we found for \(x^2 + 4 \geq 0\) was all real numbers. In interval notation, we represent this range from negative infinity to positive infinity as \( (-\backslash infty, \backslash infty)\). Parentheses are used around infinity symbols because infinity is not a specific number and cannot be included.