Problem 3 - Prealgebra

Question

Solve each inequality. Check your solution. $$-13 \geq 9+b$$

Step-by-Step Solution

Answer

The solution is \(b \leq -22\).

Step 1: Isolate the Variable

We start with the inequality:\[-13 \geq 9 + b\]To isolate the variable \(b\), we subtract 9 from both sides of the inequality. This will help us to get \(b\) by itself:\[-13 - 9 \geq b\]Simplifying the left side gives:\[-22 \geq b\] or equivalently, \(b \leq -22\).

Step 2: Check the Solution

To check if our solution is correct, substitute \(b = -22\) back into the original inequality:\[-13 \geq 9 + (-22)\]Simplifying the right side, we have:\[-13 \geq -13\]This statement is true, which means \(b = -22\) is a correct boundary solution.Now test a number less than \(-22\), such as \(b = -23\):\[-13 \geq 9 + (-23)\]This simplifies to:\[-13 \geq -14\], which is also true.Thus, our solution \(b \leq -22\) is verified.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are fundamental in mathematics. They consist of numbers, variables, and operators like addition or subtraction. In our example, we have the inequality \(-13 \geq 9 + b\). Here, 9 and \(b\) are part of the expression on the right side of inequality. Recognizing and understanding such expressions is key to solving equations and inequalities.
Algebraic expressions allow us to model real-world situations and solve various mathematical problems. By manipulating these expressions, we can find out what variables represent under different conditions and constraints.

How to Isolate the Variable

Isolating the variable is a common step in solving equations and inequalities. It means rearranging the equation so that the variable stands alone on one side. For the inequality \(-13 \geq 9 + b\), the goal is to get \(b\) by itself. To achieve this, we subtract 9 from both sides:

• \(-13 - 9 \geq b\) simplifies to \(-22 \geq b\).

Now, we have \(b\) isolated, showing \(b \leq -22\).
This step is crucial as it transforms the inequality into a form that clearly shows the relationship we are trying to solve for. Isolating the variable makes it easier to identify the range of solutions.

Importance of Solution Verification

Once we find a potential solution to an inequality, it's essential to verify it. Solution verification involves substituting the found boundary and checking if it satisfies the original inequality. In our case, by substituting \(b = -22\), we ensure the statement still holds true:

• \(-13 \geq 9 + (-22)\) results in \(-13 \geq -13\).

This equation holds, confirming that \(b = -22\) is correct. Testing further with \(b = -23\) also confirms the solution:

• \(-13 \geq 9 + (-23)\) simplifies to \(-13 \geq -14\), which is true.

Verification ensures our solution is not only mathematically valid but also makes logical sense in context.

Basics of Inequality Solving

Solving inequalities is slightly different from solving equations and involves finding a range of values that satisfy the inequality. With \(-13 \geq 9 + b\), solving involved verifying logical outcomes. When solving:

• Perform arithmetic operations on both sides to isolate the variable.
• Check the direction of the inequality; flipping it may be necessary if multiplied or divided by a negative.
• Verify solutions within the context of the original inequality.

Unlike equations, inequalities express a relationship that allows for multiple solutions rather than a single answer. Understanding and solving these can help in numerous scenarios where not just a single solution, but a range of values is needed.