Problem 38 - Prealgebra

Question

Solve each inequality. Check your solution. $$1+y \leq 2.4$$

Step-by-Step Solution

Answer

The solution is \(y \leq 1.4\).

Step 1: Isolate the Variable

The given inequality is \(1 + y \leq 2.4\). To isolate \(y\), subtract 1 from both sides of the inequality: \(y \leq 2.4 - 1\). This simplifies to \(y \leq 1.4\).

Step 2: Verify the Solution

We must check if our solution holds true. Choose a value for \(y\) that satisfies \(y \leq 1.4\). For example, let \(y = 1\). Substitute back into the original inequality: \(1 + 1 \leq 2.4\), which simplifies to \(2 \leq 2.4\). This is true, confirming the solution.

Step 3: Consider Boundary Values

Consider the boundary value for \(y = 1.4\). Substitute into the original inequality: \(1 + 1.4 \leq 2.4\), which simplifies to \(2.4 \leq 2.4\). This statement is true, confirming our boundary is correct.

Key Concepts

Isolating the Variable

When solving inequalities, the first step is often to isolate the variable. This means getting the variable alone on one side of the inequality sign. In the given exercise, the inequality is \(1 + y \leq 2.4\). The goal is to have \(y\) by itself.
To do this, we perform the same operation on both sides of the inequality. Here, we subtract 1 from both sides to eliminate the constant term on the side with \(y\). This operation looks like:

• Subtract 1 from \(1 + y\) to get \(y\)
• Subtract 1 from \(2.4\) to get \(1.4\)

After performing this action, the inequality is simplified to \(y \leq 1.4\).
Think of isolating the variable as cleaning up the equation, making it easier to see the relationship between \(y\) and the number on the other side.

Verifying the Solution

After isolating the variable and finding that \(y \leq 1.4\), it is crucial to verify that this solution is correct. Verification ensures that no mistakes were made in solving the inequality.
To verify, test a value for \(y\) that fits within the constraint \(y \leq 1.4\). A good choice here could be \(y = 1\).

• Substitute this value back into the original inequality: \(1 + 1 \leq 2.4\)
• Simplify the left side to \(2 \leq 2.4\), which is true

This true statement validates that our solution satisfies the original inequality. Try testing several values less than or equal to 1.4 to reinforce your understanding.

Understanding Boundary Values

In inequalities, a boundary value is the point where the inequality changes from true to false, or vice versa.
It's the exact value which separates the solutions from non-solutions.
For \(y \leq 1.4\), the boundary value is \(y = 1.4\).
To check this, substitute \(y = 1.4\) back into the original inequality:

• \(1 + 1.4 \leq 2.4\)
• This simplifies to \(2.4 \leq 2.4\), which is true

This confirmation shows that 1.4 is indeed part of the solution set. Boundary values are crucial for accurately understanding the limits of the inequality's solution.