Problem 27 - Beginning and Intermediate Algebra

Question

Solve each inequality. $$(k+7)^{2} \geq-9$$

Step-by-Step Solution

Answer

The inequality \((k+7)^2 \geq -9\) is always true for any value of \(k\), since the square of any real number is non-negative. Therefore, the solution is all real numbers.

Step 1: Analyzing the inequality

First, let's observe that the square of any real number is non-negative. Therefore, \((k+7)^2\) will always be greater than or equal to 0.

Step 2: Compare with given inequality

We want to determine if \((k+7)^2 \geq -9\). Since \((k+7)^2\) is always greater than or equal to 0, and 0 is greater than -9, we can conclude that this inequality is always true.

Step 3: Solution

For any value of k, the inequality \((k+7)^2 \geq -9\) will always be true. Hence, the solution to this inequality is all real numbers.

Key Concepts

Real Numbers

Real numbers include all the numbers on the number line. This encompasses both rational and irrational numbers.

• Rational numbers: These are numbers that can be expressed as a fraction, like \ \frac{1}{2} \, 4, or \ -7.\
• Irrational numbers: Numbers that cannot be expressed as a simple fraction, such as \( \pi \) and \( \sqrt{2} \).

In solving inequalities like \((k+7)^2 \geq -9\), understanding the scope of real numbers is crucial as solutions span all real numbers. Real numbers help us make sense of results that apply universally instead of just to specific cases.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operations like addition or multiplication.
The expression \((k+7)^{2}\) is a simple algebraic expression, where \(k\) is a variable, and \((k+7)\) is squared.

• Variables represent unknown values.
• Expressions can be manipulated using algebraic rules to solve problems like inequalities.

When dealing with inequalities, algebraic expressions allow us to explore different possibilities by substituting various values into the expression. This helps us understand how the inequality behaves across the number line.

Inequality Solutions

Inequalities, like equations, show the relationship between expressions. However, they tell us that one side is either greater or smaller than the other.
In our example, \((k+7)^2 \geq -9\), we are looking at a squared term that prompts us to examine the nature of square numbers.

• Square values are always non-negative.
• This means \(\forall k, (k+7)^2 \geq 0 \), and 0 is indeed greater than -9.

Thus, this tells us that the inequality holds true for all real numbers. Solving inequalities can involve recognizing these universal truths and understanding how algebraic expressions interact with inequalities to define a solution set.