Problem 18 - Algebra 2 and Trigonometry

Question

In \(13-22,\) solve each equation or inequality. Each solution is an integer. $$ 5 y-1 \geq 2 y+5 $$

Step-by-Step Solution

Answer

The solution is \(y \geq 2\).

Step 1: Move variable terms to one side

We start by simplifying the inequality by moving all terms containing the variable to one side. Subtract \(2y\) from both sides: \(5y - 2y - 1 \geq 5\)

Step 2: Simplify the expression

Next, we simplify the expression on the left side by combining like terms: \(3y - 1 \geq 5\)

Step 3: Isolate the variable term

To isolate the term containing \(y\), add \(1\) to both sides of the inequality: \(3y \geq 6\)

Step 4: Solve for the variable

Finally, divide both sides by \(3\) to solve for \(y\): \(y \geq 2\)

Key Concepts

Solving for Integer Solutions

When dealing with inequalities, especially when they result in integer solutions, it’s important to ensure all steps follow logical sequencing. The solution for inequalities in this context results in whole numbers, which are easy to plot on a number line and interpret. An integer solution in this scenario means that the solution for the variable is limited to whole numbers; it cannot be a fraction or decimal. For the problem stated, the solution was derived as an integer because when the inequality was solved, only the integer values satisfy the condition. Please remember:

• Integer solutions are attractive for clarity; they’re simple to visualize and avoid the complexities of decimals.
• Solving inequalities to find integers is fairly common in schooling maths, especially in pre-algebra and algebra. Inequalities involving integer solutions might be present in problems accessible to most middle to high school students.

The Process of Variable Isolation

Variable isolation is a core component of solving equations and inequalities. It involves manipulating the equation or inequality so that the variable is alone on one side. This is crucial for clearly understanding the value or range of values a variable can assume. In the exercise tackled, isolating the variable was essential: Subtracting terms from both sides allowed for direct simplification. Here, we subtract the smaller variable term from both sides, ensuring that the result simplifies the inequality. This makes it easier and structured to move forward with further simplifications.

• Subtract related terms on both sides proportionately; this aligns everything neatly for understanding and calculating.
• Always check your arrangement; ensuring the variable terms are kept together is significant for conveying the correct context.

Remember, variable isolation often sets the stage for swiftly determining solutions.

Simplifying the Expression

Simplifying expressions is like cleaning up a room. It involves combining like terms and simplifying parts of an equation or inequality to make the solving process easier. Throughout the steps detailed for the inequality, simplification played a major role: - Combining like terms: The initial step of rearranging terms allowed grouping of common variable terms. The goal in these scenarios is straightforward:

• Simplification reduces complexities.
• It makes it clear what operations need to be performed next.

Emphasizing on simplification ensures solutions are intelligible and provides a clear path towards the result. When dealing with equations, it couldn't be more essential. Every expression simplified brings the solution a step closer to the conclusion.