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Inequalities in algebra are expressions that show the relationship between two values, indicating that one value is less than, greater than, or possibly equal to another. Solving an inequality means finding all possible values that satisfy this condition. The process is fairly similar to solving regular algebraic equations.

One of the primary steps is to isolate the variable. In our example, we have the inequality: \[ 9(y - 4) \leq 36 \]To isolate the term with the variable, divide every term by 9, yielding:\[ y - 4 \leq 4 \]

Next, get the variable by itself. Add 4 to both sides to obtain:\[ y \leq 8 \]The solution to the inequality indicates that all values of \( y \) up to and including 8 satisfy the inequality.

After solving an inequality, it's helpful to graph the solution to visually understand the range of possible values. This makes the solution more intuitive.

Start by drawing a number line and labeling it with relevant numbers. In the example, mark at least 0 and 8 on the number line. The endpoint of 8 should have a filled circle, which tells us that 8 is part of the solution (since \( y \) can be equal to 8 as indicated by the "\( \leq \)" symbol).

Then, shade the area on the number line to the left of 8, showing that all numbers less than 8 also satisfy the inequality. This graphical method helps in quickly seeing the solution set for the inequality.

Elementary algebra is the foundation of all algebraic operations, dealing with symbols and letters representing numbers. These basic principles help in solving equations and inequalities effectively.

To approach exercises like inequalities in elementary algebra, understanding operations like:

• Adding or subtracting the same number on both sides
• Multiplying or dividing both sides by a positive number

is crucial. A key point is that when multiplying or dividing by a negative number, one must reverse the inequality sign, although our specific exercise doesn't involve such a change.

This strong base ensures that when you encounter more complex problems, the principles and techniques used in elementary algebra equip you for success. It's about getting comfortable with manipulating equations to find solutions that fit the given conditions.