Problem 28 - Algebra 1: Concepts and Skills

Question

Check to see if the given value of the variable is or is not a solution of the inequality. $$ 4+k \geq 32 ; k=30 $$

Step-by-Step Solution

Answer

Yes, the given value of the variable \(k = 30\) is a solution of the inequality \(4 + k ≥ 32\).

Step 1: Understand the Problem

In this problem, the inequality given is \(4 + k ≥ 32\), where \(k = 30\). It is needed to substitute the value of \(k\) into the inequality and check if it holds true.

Step 2: Substitute the Value of k

Replace the \(k\) in the inequality \(4 + k ≥ 32\) with \(30\), to get the inequality \(4 + 30 ≥ 32\)

Step 3: Evaluate the Inequality

Simplify the left-side of the inequality to get \(34 ≥ 32\)

Step 4: Check the Inequality

Since \(34\) is indeed greater than \(32\), the inequality \(34 ≥ 32\) holds as true.

Key Concepts

Substitution in Inequalities

Substitution in inequalities is a fundamental concept in mathematics. It allows you to determine if a specific value satisfies an inequality.

When you substitute, you replace a variable with a number. This transforms the inequality into a simpler arithmetic expression. For example, in the inequality \(4 + k \geq 32\), if we want to check if \(k = 30\) is a solution, we replace \(k\) with \(30\).

• This turns our inequality into \(4 + 30 \geq 32\).
• Substitution is often the first step you take when beginning to analyze an inequality.
• It helps to verify whether a chosen number meets the condition set by the inequality.

Make sure you're clear about what value you're substituting and where it needs to be in the inequality. With the correct substitution, the inequality becomes much easier to evaluate.

Evaluating Inequalities

Evaluating inequalities involves simplifying both sides of the inequality to make a comparison. After substituting the given value, simplify it step-by-step.

Using the inequality \(4 + 30 \geq 32\), you simplify as follows:

• First, add \(4\) and \(30\), which equals \(34\).
• Now, the inequality reads \(34 \geq 32\).

Upon simplifying, you compare the two sides of the inequality.

• If your simplification results in a true statement, like \(34\) being greater than or equal to \(32\), the original inequality with the substituted value is valid.
• Otherwise, you'd conclude that the value doesn't satisfy the inequality.

Evaluation is crucial as it confirms the validity of your substitution and checks mathematical correctness.

Solving Inequalities

Solving inequalities means finding all possible values that make an inequality true, although in some cases like this specific exercise, we're only checking one value.

To solve inequalities systematically:

• First, express the inequality clearly and include any substitutions required.
• Then, simplify the inequality as much as possible to expose the relationship between numbers.
• Finally, analyze the results to determine which values satisfy the inequality.

Though not always necessary for single value checks, understanding solving methods is valuable for tackling broader inequality problems. These methods can range from graphing solutions to refining algebraic expressions, depending on complexity. Remember, solving inequalities expands beyond looking at one specific value to understanding a whole range of potential solutions.