Solve the Inequality for p 4p< 32

Solution:

Step 1:

Execute division on both sides of the inequality $4p<32$ by $4$. Obtain $\frac{4p}{4}<\frac{32}{4}$.

Step 2:

Reduce the expression on the left-hand side.

Step 2.1:

Eliminate the common factor of $4$.

Step 2.1.1:

Remove the common factor. Write $\frac{\cancel{4}p}{\cancel{4}}<\frac{32}{4}$.

Step 2.1.2:

Express $p$ over $1$. Thus, we have $p<\frac{32}{4}$, which simplifies to $p<\frac{32}{4}$, and finally to $p<\frac{32}{4}$.

Step 3:

Condense the expression on the right-hand side.

Step 3.1:

Compute the division of $32$ by $4$. This yields $p<8$.

Step 4:

Present the solution in various acceptable formats.

Inequality Representation: $p<8$

Interval Notation: $(-\mathrm{\infty },8)$

Knowledge Notes:

The process of solving a linear inequality is similar to solving a linear equation, with the primary difference being the inequality sign. Here are the relevant knowledge points:

1. Division Property of Inequality: When both sides of an inequality are divided by a positive number, the direction of the inequality remains the same. In this case, dividing by $4$ does not change the direction of the inequality.

2. Simplification: After dividing, it is important to simplify the expression to find the solution to the inequality. Simplification involves canceling out common factors and performing arithmetic operations.

3. Interval Notation: This is a way of writing sets of numbers, often used to describe the solution set of an inequality. For the inequality $p<8$, the interval notation is $(-\mathrm{\infty },8)$, which means that $p$ can be any number less than $8$.

4. Inequality Representation: The solution to an inequality can be represented in multiple ways, including inequality form (e.g., $p<8$) and interval notation.

5. Latex Formatting: In the solution, Latex is used to format mathematical expressions, ensuring that they are clearly and correctly presented. For example, $\frac{4p}{4}<\frac{32}{4}$ is rendered in Latex to display the fraction and inequality properly.