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Inequalities describe a relationship between two values where they are not equal. They show that one value is larger or smaller than the other.
Inequalities use symbols like \( \leq \) (less than or equal to) and \( \geq \) (greater than or equal to).
These symbols indicate how one value compares to another.
For example, in the inequality \(-\frac{3}{4} \leq x\), the symbol \( \leq \) means that \(x\) can be \(-\frac{3}{4}\) or any number greater than \(-\frac{3}{4}\).
When we express inequalities in algebra, we often need to convert them into different forms for better understanding.
One common method is interval notation, which provides a concise way to express a range of values that satisfy the inequality.
By learning to write inequalities properly and understanding their symbols, students can solve problems more effectively and understand mathematical concepts deeply.

Number line graphing is a visual way to represent inequalities and intervals.
A number line graph helps us see the range of values that satisfy an inequality.
To graph an interval, follow these simple steps:

• Draw a horizontal line and mark numbers on it.
• Identify the key points given in the interval. For \([-\frac{3}{4}, \infty)\), the key point is \(-\frac{3}{4}\).
• Place a circle on this key point. Use a closed circle if the value is included (as with the \(\text{-}\frac{3}{4}\) in our case), or an open circle if it is not.
• Shade the region that represents all possible values based on the inequality. For \([-\frac{3}{4}, \infty)\), shade to the right of \(-\frac{3}{4}\) as it includes all numbers greater than or equal to \(-\frac{3}{4}\).

This visual representation can make it easier to understand and solve inequalities.
It shows us exactly which values are included in the solution set and which are not.

Interval notation is a shorthand way to write the range of values that satisfy an inequality.
It shows where the value starts and ends, including or excluding endpoints.
Here’s how it works:

• Use brackets \([]\) when the endpoint is included, like in \([-\frac{3}{4}, \infty)\).
• Use parentheses \( () \) when the endpoint is not included.
• For the interval \([-\frac{3}{4}, \infty)\), \infty always uses a parenthesis because infinity is not a number we can reach.

Let’s apply interval notation to our example:
The inequality \( -\frac{3}{4} \leq x \) means all numbers greater than or equal to \(-\frac{3}{4}\).
In interval notation, we write this as \([-\frac{3}{4}, \infty)\).
This compact notation helps communicate the solution clearly and efficiently.
Mastering interval notation is vital for solving and graphing inequalities elegantly.