Question

The inequality \(x^{2}+1<-5\) has no real solution. Explain why.

Short answer

The inequality \(x^{2}+1<-5\) has no real solution because a non-negative number cannot be less than a negative number.

Step-by-step solution

- Analyze the Inequality

First, consider the given inequality: \(x^{2}+1<-5\). We need to determine if any real values of \(x\) exist that satisfy this inequality.

- Simplify the Inequality

Rewrite the inequality to make it easier to work with: \(x^{2} < -6\).

- Examine the Left Side

The term \(x^{2}\) represents a squared term. For all real numbers \(x\), \(x^{2}\) is always greater than or equal to zero because squaring any real number cannot result in a negative number.

- Compare with the Right Side

The inequality we need to solve is \(x^{2} < -6\). However, the left side \(x^{2}\) is always non-negative, while the right side \(-6\) is a negative number.

- Conclude No Real Solution

Since a non-negative number \(x^{2}\) can never be less than a negative number such as \(-6\), we conclude that there are no real values of \(x\) that satisfy the inequality. Therefore, the original inequality has no real solution.

Key concepts

Real Numbers

Real numbers are the set of all numbers that can be found on the number line. This includes both rational numbers, such as integers and fractions, and irrational numbers that cannot be expressed as fractions, such as \(\pi\) and square roots of non-perfect squares. Real numbers can be positive, negative, or zero. Essentially, real numbers encompass every possible number without imaginary components or complex numbers. Any number you encounter in everyday life, whether it be temperature, age, or distance, is a real number. When solving equations or inequalities in algebra, we usually work within the realm of real numbers to find solutions that make practical sense.

Quadratic Inequalities

Quadratic inequalities involve a quadratic expression and inequality signs like \(>\), \( <\), \(\geq\), or \( \leq\). A quadratic expression is typically in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Quadratic inequalities can be tricky because they may have multiple solutions or require checking intervals on the number line.
To solve a quadratic inequality, follow these steps:

• First, rewrite the inequality to have zero on one side.
• Factor the quadratic expression, if possible.
• Determine the critical points by setting the quadratic equation equal to zero.
• Test intervals around the critical points to see which intervals satisfy the inequality.

Understanding whether the quadratic expression is greater than or less than zero will guide you in identifying the solution set.

Non-Negative Numbers

Non-negative numbers include all real numbers that are greater than or equal to zero. This means they include all positive numbers and zero itself. Non-negative numbers do not include any negative values.
The concept of non-negative numbers is essential in many areas of math because certain operations or functions cannot yield negative results. For instance, the square of any real number is always non-negative. When working within algebra or calculus, ensuring results are non-negative helps maintain the logical consistency of solutions.
In our original exercise, the term \(x^2\) is non-negative because it represents the square of a real number, and squaring any real number cannot produce a negative value. Understanding this principle helps us conclude whether certain inequalities are solvable within the realm of real numbers.

Solving Inequalities

Solving inequalities involves determining the range of values that satisfy the given relation. Unlike equations, inequalities can have a range of solutions instead of a single value. An important step in solving inequalities is to understand the direction of the inequality sign and whether it is strict (\(>\) or \( <\)) or inclusive (\(\geq\) or \( \leq\)).

• First, isolate the variable on one side of the inequality.
• Perform algebraic operations similar to solving an equation, but remember to reverse the inequality sign if you multiply or divide by a negative number.
• Graph the solution on a number line to visualize the range of values that satisfy the inequality.

In the context of quadratic inequalities, it is necessary to find the critical points where the expression equals zero and then determine which intervals satisfy the inequality by testing points within those intervals.
In the provided exercise, we concluded that there are no real solutions to \(x^2 + 1 < -5\) because \(x^2\) is always non-negative and can never be less than a negative number.