Problem 55 - Algebra 1

Question

Check whether the number is a solution of the equation or the inequality. \(32-5 y>11 ; 3\)

Step-by-Step Solution

Answer

Yes, the number 3 is a solution to the inequality \(32 - 5y > 11\)

Step 1: Identify the inequality and the given number

The inequality given is \(32-5y > 11\) and the given number, which is presumed to be the value of \(y\), is \(3\).

Step 2: Substitute the number into inequality

Put \(3\) in place of \(y\) in the inequality. This will alter the inequality to: \(32 - 5(3) > 11\).

Step 3: Simplify and solve the inequality

Now, simplify and solve the inequality. This would translate the inequality to: \(32 - 15 > 11\) which in turn simplifies to \(17 > 11\). This statement is true, which signifies the number 3 is a solution to the inequality \(32 - 5y > 11\).

Key Concepts

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and mathematical operations. Imagine them as phrases formed by combining math words with no equal sign. Expressions can exist in different forms like monomials, binomials, and polynomials.

• Monomials have one term, such as \(5y\).
• Binomials have two terms, like \(32 - 5y\).
• Polynomials contain more than two terms.

In the exercise, the expression \(32 - 5y\) shows subtraction and multiplication combined. This expression becomes part of an inequality, forming a statement that tells us something about the relationship between the expression and another value. Understanding how to manipulate and evaluate these expressions is a key skill in solving algebraic problems.

Solving Inequalities

An inequality expresses a comparison, like greater than (\(>\)) or less than (\(<\)). Solving inequalities involves finding the range of values for which the inequality holds true. In this exercise, we focus on one specific case of testing if a given number fits into the range.
You start by rewriting the inequality with the given number in place of the variable. Then, you simplify the expression and check if the resulting statement is true.

• Rewrite the inequality by substituting the known value.
• Simplify the calculations.
• Evaluate if the statement is true, false, or needs further analysis in complex situations.

True inequalities identify possible solutions, while false ones show what numbers don't work. Practicing these steps is crucial for mastering inequalities.

Substitution Method

The substitution method involves replacing a variable with a given value to simplify an expression or inequality. This technique is very handy when you want to check if a particular solution is valid for an equation or inequality.
In our context, you substitute \(y = 3\) into the inequality \(32 - 5y > 11\). By placing the number into the expression, you simplify and calculate to see if the original condition holds.

• Identify the variable to substitute.
• Replace the variable in the expression or inequality with the given number.
• Simplify the resulting expression or inequality for a conclusion.

This straightforward approach helps verify solutions quickly and is particularly useful in linear equations and inequalities.