Problem 19 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ y+19 \geq 2 $$

Step-by-Step Solution

Answer

Question: Solve the inequality and find the possible values of y: $$y + 19 \geq 2$$ Answer: The possible values of y are all values greater than or equal to $$-17$$, or $$y \geq -17$$.

Step 1: Identify the inequality

We need to solve the inequality $$y + 19 \geq 2$$ for the variable y.

Step 2: Isolate y

To isolate y, we need to subtract 19 from both sides of the inequality. This gives us: $$y \geq 2 - 19$$

Step 3: Simplify the inequality

Now, subtract 19 from 2 to simplify the inequality: $$y \geq -17$$

Step 4: Write the solution

The solution to the inequality is $$y \geq -17$$. This means that all values of y greater than or equal to -17 satisfy this inequality.

Key Concepts

Algebraic Inequalities

When tackling algebraic inequalities, students are often challenged with the concept that, unlike equations, these inequalities don't suggest equality but rather a range of possible solutions. An inequality can indicate that one side is greater than, less than, greater than or equal to, or less than or equal to the other side.

In the exercise provided, we are given the inequality: \( y + 19 \text{≥} 2 \). Understanding that this means 'y plus 19 is greater than or equal to 2' is crucial. Algebraic inequalities like this one often represent real-world scenarios where there are multiple acceptable solutions, such as determining the minimum height for a ride at an amusement park or calculating a budget limit.

Isolate the Variable

One of the most important skills in solving algebraic inequalities is learning how to isolate the variable. 'Isolating the variable' means manipulating the inequality such that the variable you’re solving for is alone on one side, and everything else is on the other side.

To isolate the variable in our example, we perform operations on both sides of the inequality in a symmetrical fashion to maintain the balance. We subtract 19 from both sides to effectively 'move' the 19 across and get: \( y \text{≥} -17 \). It's important to remember that whatever operation you do to one side—whether it be addition, subtraction, multiplication, or division—it must also be done to the other side to keep the inequality true.

Inequality Solution

Identifying the solution set for an inequality is the final step in the process. Once the variable is isolated, as we've done previously with \( y \text{≥} -17 \), we interpret this solution within the context it may represent.

In essence, the solution to an inequality is not just one number but an entire set of numbers that satisfy the condition established by the inequality. In this case, any number greater than or equal to -17 is part of the solution set. When graphing, we would shade all points to the right of -17 on the number line and include a closed dot at -17, to indicate that -17 is included in the solution set. This visual representation helps solidify the concept that an infinite number of values can fulfill the condition of an inequality.