Problem 55 - Algebra 1: Concepts and Skills

Question

CHECKING SOLUTIONS OF INEQUALITIES Check to see if the given value of the variable is or is not a solution of the inequality. $$ 7 g \geq 47 ; g=7 $$

Step-by-Step Solution

Answer

The given value \(g = 7\) is a solution to the inequality \(7g \geq 47\).

Step 1: Substitute the Given Value for \(g\)

Substitute the given value of \(g = 7\) into the inequality to check if the resulting expression is true. By substituting, the inequality becomes \(7*7 \geq 47\).

Step 2: Evaluate the Expression

Evaluate the expression on the left side of the inequality, which is \(7*7\). The result is 49.

Step 3: Check the Inequality

Now, with the left hand side evaluated, the inequality appears as \(49 \geq 47\). It's visible that 49 is indeed greater than or equal to 47, so the inequality is true.

Key Concepts

Evaluating Expressions

In mathematics, evaluating expressions involves finding the value of an expression when the values of the variables are substituted. In the context of inequalities, it means computing the result on one or both sides to determine if the inequality holds. For example, consider the inequality \(7g \geq 47\). When we evaluate the expression \(7g\) for \(g = 7\), we perform the multiplication: \(7 \times 7\), which equals 49.

By evaluating this expression, we know the left side of the inequality, giving us a clear numerical value to use in comparison with the right side. This process is crucial as it transforms abstract expressions into concrete numbers that can easily be compared.

Substitution Method

The substitution method is a powerful technique used in algebra, especially when dealing with equations and inequalities. It involves replacing a variable with its given value to simplify the expression or inequality. In this exercise, we'll take the example \(7g \geq 47\), and use substitution to analyze it.

The substitution step looks like this: wherever \(g\) appears, we replace it with 7, rendering the expression \(7 \times 7 \geq 47\). This helps us check if the inequality holds true for the specific value of \(g\).

Benefits of substitution method include:

• Simplifies complex expressions into manageable calculations.
• Helps in verifying solutions by providing specific values.
• Makes it easier to test different scenarios or values quickly.

When evaluating inequalities, proper substitution is vital in ensuring that you're solving the problem accurately.

Mathematical Reasoning

Mathematical reasoning is the logical thought process needed to deduce whether mathematical statements, like equations or inequalities, are true or false. It involves analyzing expressions and making judgments based on mathematical principles.

For instance, after substituting and evaluating \(7 \times 7\) in our inequality \(7g \geq 47\), we arrive at 49 \(\geq\) 47. Now it's time to use mathematical reasoning to interpret this.Here is how you use mathematical reasoning:

• Recognize that 49 is indeed greater than 47, confirming the inequality is true.
• Understand why the inequality holds by recalling that \(\geq\) means "greater than or equal to," covering both conditions.
• Develop a logical argument or explanation on why the findings are correct.

Strong mathematical reasoning enhances problem-solving skills, ensuring solutions aren't based on guessing but sound logic and critical thinking.