Solve the Inequality for x x+7< 0

Solution:

Step 1:

Isolate the variable $x$ by deducting $7$ from each side of the inequality to obtain $x < -7$.

Step 2:

Express the solution in different ways. In inequality notation, it is $x < -7$. In interval notation, the solution is $(-\infty, -7)$.

Knowledge Notes:

To solve a linear inequality, such as $x + 7 < 0$, the goal is to isolate the variable on one side of the inequality sign. This process is similar to solving a linear equation, with the key difference being the direction of the inequality sign, which must be preserved throughout the process.

Here are the relevant knowledge points and detailed explanations:

1. Subtraction Property of Inequality: This property allows us to subtract the same number from both sides of the inequality without changing its direction. In this case, subtracting $7$ from both sides gives us $x < -7$.

2. Inequality Notation: The inequality notation simply uses the inequality symbols (e.g., $<$, $>$, $\leq$, $\geq$) to show the relationship between the variable and a number. Here, $x < -7$ means that $x$ is any number less than $-7$.

3. Interval Notation: Interval notation is a way of representing a set of numbers between two endpoints. The parentheses, $( )$, indicate that the endpoints are not included in the set, while brackets, $[ ]$, would indicate that the endpoints are included. The solution to the inequality $x < -7$ in interval notation is $(-\infty, -7)$, which represents all numbers less than $-7$.

4. Inequalities and the Number Line: When representing inequalities on a number line, an open circle is used to represent that an endpoint is not included in the solution set (corresponding to a parenthesis in interval notation), while a closed circle indicates that the endpoint is included (corresponding to a bracket in interval notation).

5. Linear Inequalities: A linear inequality looks much like a linear equation, except it has an inequality sign instead of an equality sign. The solutions to a linear inequality are a range of values rather than a single number.

In summary, solving linear inequalities involves using properties of inequalities to isolate the variable and then representing the solution in various forms, such as inequality notation or interval notation. It's important to remember that the direction of the inequality sign must remain consistent unless you multiply or divide by a negative number, in which case the inequality sign reverses.