Problem 16 - Prealgebra

Question

Solve each inequality. Check your answer. $$3 \geq-2+y$$

Step-by-Step Solution

Answer

The solution is \( y \leq 5 \).

Step 1: Isolate the Variable

To solve for \(y\), we need to isolate it on one side of the inequality. We start with the given inequality: \[ 3 \geq -2 + y \] Add \(2\) to both sides to remove \(-2\) from the right side:\[ 3 + 2 \geq y \] Simplify the left side:\[ 5 \geq y \] which can also be written as \( y \leq 5 \).

Step 2: Write the Result

The inequality \( y \leq 5 \) means that \(y\) can take any value that is less than or equal to \(5\).

Step 3: Verify the Solution

To verify the solution \( y \leq 5 \), we can test a value of \(y\) that is less than or equal to \(5\) in the original inequality. Let's test \(y = 4\):Substitute into \(3 \geq -2 + y\):\[ 3 \geq -2 + 4 \] Calculate the right side:\[ 3 \geq 2 \] This is true. Now, let's test the upper bound \(y = 5\):\[ 3 \geq -2 + 5 \]Calculate the right side:\[ 3 \geq 3 \] This is true as well, confirming \(y \leq 5\) is correct.

Key Concepts

Prealgebra

Prealgebra is the foundational step toward learning algebra. It introduces students to basic mathematical concepts and operations that will be used more extensively in algebra, such as arithmetic operations, fractions, decimals, and simple equations. One common focus is on understanding inequalities, which are statements that show how one value compares to another.

In inequalities, symbols like \( \leq \) (less than or equal to) and \( \geq \) (greater than or equal to) are used to compare values. This exercise illustrates these concepts with simple arithmetic and shows how inequalities expand upon the idea of equations by allowing for a range of possible solutions rather than a single answer. Understanding these concepts is essential for success in algebra and other higher-level mathematics.

Isolate the Variable

Isolating the variable is a critical step in solving equations and inequalities. The goal is to get the variable alone on one side, making it easy to find its value. This often involves using inverse operations, such as adding the opposite number to both sides or dividing by the coefficient of the variable.

In this exercise, the original inequality is \( 3 \geq -2 + y \). The solution starts by isolating \( y \), which requires moving the \(-2\) to the other side. We do this by adding 2 to both sides of the inequality:

• Add 2 to both sides: \( 3 + 2 \geq y \)
• Simplify, yielding \( 5 \geq y \) or equivalently \( y \leq 5 \)

By isolating \( y \), we determine the range of values that satisfy the inequality.

Verify Solution

Verifying the solution of an inequality ensures that the manipulated inequality is correct. This involves selecting numbers within the solution set and substituting them back into the original inequality.

For the inequality \( y \leq 5 \), we substitute values like \( y = 4 \) and \( y = 5 \) to check if they satisfy the original inequality \( 3 \geq -2 + y \).

• Test \( y = 4 \): Substitute to get \( 3 \geq -2 + 4 \), which simplifies to \( 3 \geq 2 \). This statement is true.
• Test \( y = 5 \): Substitute to get \( 3 \geq -2 + 5 \), simplifying to \( 3 \geq 3 \). This is also true.

By confirming these, we establish that the solution \( y \leq 5 \) is valid, demonstrating our solution steps were accurate.