Problem 32 - 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. $$ y-33 \geq 51 ; y=84 $$

Step-by-Step Solution

Answer

Yes, the given value for 'y' (84) is a solution to the inequality

Step 1: Identify the given inequality and the variable's value

The given inequality is \(y-33 \geq 51\) and the value of the variable 'y' given to be checked is 84

Step 2: Substitute the given value of the variable in the inequality

Replace 'y' in the inequality with the given value. It then becomes: \(84 - 33 \geq 51\)

Step 3: Perform the calculation

Calculate the left side of the inequality, \(84 - 33\), which equals 51. So the inequality now is \(51 \geq 51\)

Step 4: Check the validity of the inequality

The inequality \(51 \geq 51\) holds true as 51 is indeed equal to 51

Key Concepts

Inequality Solutions

Inequalities are mathematical expressions that compare two values, often using signs like <, >, ≤, or ≥. Solving inequalities is about finding all the possible values that make the inequality true. Unlike equations, inequality solutions can often range over a set of numbers rather than being a single value.

Let's consider a straightforward example. If you were to solve the inequality \(x + 5 > 10\), you would subtract 5 from both sides to isolate \(x\), resulting in \(x > 5\). In this case, any number greater than 5 would be a solution. It's essential to remember that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign to maintain the correct relationship between the two sides.

Substitution Method

The substitution method in algebra is a way of finding out whether a particular value is a solution to an inequality or equation. This involves replacing the variable in the inequality with the given number and then checking if the statement is true.

For instance, in our exercise, the inequality \(y-33 \geq 51\) is provided along with \(y=84\). By substituting 84 in place of \(y\), you get \(84-33 \geq 51\). After performing the subtraction, you end up with \(51 \geq 51\), which confirms that \(y=84\) is indeed a solution since both sides are equal, satisfying the \(\geq\) condition. This methodical approach of substitution helps clearly validate or invalidate potential solutions to inequalities.

Algebraic Reasoning

Algebraic reasoning involves the process of thinking logically about relationships between variables and numbers to solve problems. It's an essential skill in algebra, helping to identify patterns, make conjectures, and even justify conclusions.

In the context of our exercise, algebraic reasoning helps us to understand why after substituting \(y=84\) into the inequality \(y-33 \geq 51\), and obtaining \(51 \geq 51\), we can conclude that the inequality holds true. This is because algebraically, we recognize that the \(\geq\) sign means 'greater than or equal to'. Since 51 is equal to 51, the statement is accurate. Thus, through algebraic reasoning, we can generalize and understand that any time we find the two sides of an inequality to be the same number, and the inequality includes equality (like \(\geq\) or \(\leq\)), the proposed value is indeed a valid solution.