Solve the Inequality for x y> x+3

Solution:

Step 1:

Position $x$ to the left in the inequality to get $x + 3 < y$.

Step 2:

Isolate $x$ by deducting $3$ from each side, resulting in $x < y - 3$.

Knowledge Notes:

The problem at hand involves solving an inequality. Inequalities are mathematical expressions involving the symbols $>$ (greater than), $<$ (less than), $\geq$ (greater than or equal to), and $\leq$ (less than or equal to). They show the relationship between two values or expressions. The process of solving an inequality is to find the set of all possible values of the variable that make the inequality true.

Here are the relevant knowledge points and explanations for solving the inequality $y > x + 3$:

1. Rearranging Inequalities: Similar to equations, inequalities can be rearranged to isolate the variable of interest. However, one must be careful to preserve the direction of the inequality when moving terms from one side to the other.

2. Subtracting from Both Sides: When solving inequalities, you can subtract the same number from both sides without changing the inequality's direction. This is what is done in Step 2 of the solution process.

3. Maintaining the Inequality Direction: Unlike multiplication or division by negative numbers, adding or subtracting does not change the direction of the inequality. If you multiply or divide both sides of an inequality by a negative number, the inequality's direction must be reversed.

4. Solution Sets: The solution to an inequality is not a single number but a set of numbers that satisfy the inequality. In this case, the solution set includes all real numbers $x$ that are less than $y - 3$.

5. Graphical Representation: Inequalities can be represented on a number line, where the solution set is often indicated by a shaded region.

6. Checking Solutions: It is always a good practice to check if a particular value satisfies the inequality. For example, if $y = 5$, then according to the solution $x < 2$, any $x$ value less than 2 should satisfy the original inequality $y > x + 3$.

The solution process provided maintains the format of the original problem-solving process, including the use of markdown for titles and the step-by-step approach. The mathematical expressions are rendered in LaTeX format to ensure clarity and proper formatting.