Problem 19 - Algebra 2 and Trigonometry

Question

In \(13-22,\) solve each equation or inequality. Each solution is an integer. $$ 9 y+2 \leq 7 y $$

Step-by-Step Solution

Answer

The solution is \(y = -1\).

Step 1: Bring All Variables to One Side

We start by bringing all the terms involving the variable to one side of the inequality. To do this, subtract \(7y\) from both sides of the inequality:\[9y + 2 - 7y \leq \ 7y - 7y\]This simplification gives us:\[2y + 2 \leq 0\]

Step 2: Isolate the Variable Term

Next, we need to isolate the term with the variable \(y\). To do this, subtract 2 from both sides of the inequality:\[2y + 2 - 2 \leq 0 - 2\]This simplifies to:\[2y \leq -2\]

Step 3: Solve for the Variable

Now, we solve for \(y\) by dividing both sides of the inequality by 2:\[\frac{2y}{2} \leq \frac{-2}{2}\]This results in the simplified inequality:\[y \leq -1\]

Step 4: Interpret the Solution

The inequality \(y \leq -1\) tells us that the solution for \(y\) is any integer less than or equal to \(-1\). The integer solution to this inequality is \(y = -1\).

Key Concepts

Integer Solutions

Integer solutions to an inequality or equation are values that satisfy the condition expressed using whole numbers. These include positive numbers, negative numbers, and zero, without any fractions or decimals. In our exercise, we need to find integer solutions where the variable satisfies a certain condition.

For the inequality given, namely \(9y + 2 \leq 7y\), the solution needs to be expressed in terms of integers. When we solve the inequality for \(y\), we find that \(y \leq -1\). This means that the smallest integer,

• equal to or satisfying the inequality condition,
• is \(-1\).

Of course, because integers extend infinitely in both positive and negative directions, there are many integer solutions. However, the main idea is to identify the smallest integer and all integers less than this smallest integer that satisfy the inequality condition.

By identifying integer solutions, students can ensure the precision of their answers, especially important in situations where decimals and fractions won’t suffice.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving equations and inequalities. This technique involves rearranging and simplifying expressions to isolate the variable of interest. The goal is to make the inequality or equation easier to understand and solve.

In the provided inequality solution, algebraic manipulation is used to bring all variable terms to one side of the inequality. We start by subtracting \(7y\) from both sides, leading to the expression \(2y + 2 \leq 0\). This step is pivotal as it consolidates like terms and simplifies the expression.

Isolating the Variable

Isolating the variable is about getting the variable \(y\) by itself on one side of the inequality. In many cases, this involves several operations such as addition, subtraction, multiplication, or division. For example:

• We subtract 2 from both sides, resulting in \(2y \leq -2\).
• Then, we divide every term by 2, to finally solve for \(y\).

Each step has its own rationale:

• Subtraction helps to clear constants from one side.
• Division is used to adjust the coefficient of the variable to one.

These manipulations must maintain the equality or inequality, ensuring that each transformation is valid and logical.

Inequality Solving Steps

Solving inequalities involves a systematic approach similar to equation-solving, but with key differences, especially when multiplying or dividing by negative numbers because the inequality sign reverses in these operations.

Step-by-Step Breakdown

To tackle the inequality \(9y + 2 \leq 7y\) effectively, follow these structured steps:

• **Combine like terms:** Start by moving all terms with the variable \(y\) to one side, which involves subtracting \(7y\) from each side, giving us \(2y + 2 \leq 0\).
• **Isolate the variable term:** To clear any additional constants that accompany the variable term, subtract 2 from both sides, resulting in the simpler inequality \(2y \leq -2\).
• **Solve for the variable:** Divide every term by 2 to make the coefficient of \(y\) equal to 1, leading to the inequality \(y \leq -1\).

This solution delineates all integer values that \(y\) can assume while upholding the inequality’s conditions.Remember, the solving steps must be performed carefully to respect the principles of inequalities. Each step requires careful attention to ensure that transformations accurately reflect the inequality, preserving its original meaning and constraints.