Solve for B 18+0.08B> =75

Solution:

Step 1: Isolate the variable $B$ on one side of the inequality.

• Step 1.1: Subtract $18$ from both sides to remove the constant term from the left side.

  $$ 0.08B \geq 75 - 18 $$

• Step 1.2: Perform the subtraction to find the new inequality.

  $$ 0.08B \geq 57 $$

Step 2: Solve for $B$ by eliminating the coefficient.

• Step 2.1: Divide the inequality by $0.08$ to isolate $B$.

  $$ \frac{0.08B}{0.08} \geq \frac{57}{0.08} $$

• Step 2.2: Simplify the inequality.

  • Step 2.2.1: Remove the common factor of $0.08$.

    • Step 2.2.1.1: Cancel out $0.08$.

      $$ \frac{\cancel{0.08}B}{\cancel{0.08}} \geq \frac{57}{0.08} $$

    • Step 2.2.1.2: Recognize that $B$ divided by $1$ is just $B$.

      $$ B \geq \frac{57}{0.08} $$

• Step 2.3: Calculate the numerical value.

  • Step 2.3.1: Divide $57$ by $0.08$ to find the value of $B$.

    $$ B \geq 712.5 $$

Step 3: Express the solution in different forms.

• Inequality Form: $$ B \geq 712.5 $$
• Interval Notation: $$ [712.5, \infty) $$

Knowledge Notes:

The problem at hand is an inequality that involves solving for the variable $B$. The process of solving inequalities is similar to solving equations, with the added consideration that multiplying or dividing by a negative number will reverse the inequality sign.

Here are the relevant knowledge points:

1. Isolating the variable: In any equation or inequality, the first step is usually to isolate the variable on one side. This is done by performing the same operation on both sides of the inequality.

2. Simplifying the inequality: After isolating the variable, the next step is to simplify the inequality by canceling out common factors and performing arithmetic operations.

3. Division by a positive number: When dividing both sides of an inequality by a positive number, the direction of the inequality remains the same.

4. Solution representation: The solution to an inequality can be represented in various forms, including inequality notation (e.g., $B \geq 712.5$) and interval notation (e.g., $[712.5, \infty)$).

5. Interval notation: This notation is used to describe the set of numbers that satisfy the inequality. The square bracket indicates that the number is included in the set, while a parenthesis would indicate that the number is not included.

6. Inequalities with variables: When solving inequalities with variables, the goal is to find the range of values that the variable can take that satisfies the inequality.

In LaTeX, the use of $\geq$represents the "greater than or equal to" symbol (鈮?, and $\cancel$is used to visually indicate the cancellation of terms. The use of $\frac$is for fractions, where the numerator and denominator are placed within the curly braces ${}$following $\frac$. Interval notation is expressed using brackets $[]$for inclusive bounds and parentheses $()$for exclusive bounds.