91Ó°ÊÓ

Interval notation is a concise way of expressing a range of values that satisfy an inequality. It uses brackets and parentheses to show whether endpoints are included or excluded. In the exercise provided:

• The solution of the inequality is expressed as \([1, \infty)\).
• A square bracket \([\ ]\) is used when an endpoint is included in the set. Here, the number 1 is included, as the inequality allows \(x\) to be exactly 1.
• A parenthesis \((\ )\) is used when an endpoint is not included, which applies when the endpoint is infinity or negative infinity since these are not reachable values.

This notation helps give a complete "picture" of what values make the original inequality true.
Reading it, the interval \([1, \infty)\) means any real number greater than or equal to 1.

Inequalities are mathematical expressions involving the symbols \(<\), \(>\), \(\leq\), and \(\geq\). To solve an inequality means to find all the values of the variable that make the inequality true. This process is slightly different from solving equations as inequalities convey a range of solutions rather than a single solution.
When solving, you can apply similar steps as with equations, such as:

• Adding or subtracting the same value from both sides to simplify the inequality.
• Multiplying or dividing both sides by a positive number—as long as you remember that multiplying or dividing by a negative number reverses the inequality sign.

The important takeaway is that the rules that keep the inequality balanced need to be strictly followed to ensure a correct solution. In the exercise, subtraction was used to isolate the variable \(x\).
This gives us the inequality \(x \geq 1\), showing that all numbers 1 or greater are solutions.

Linear inequalities are similar to linear equations but instead of equality, they express a relationship using inequalities. They always use a linear expression involving a variable and a constant.
Linear inequalities can have one variable like in our exercise:

• Where the inequality \( 7x - 3 \geq 6x - 2 \) was simplified to \(x \geq 1\).

The graph of a linear inequality on a number line or coordinate plane includes a range of values that satisfies the inequality. For the inequality \(x \geq 1\), you would shade all numbers to the right of 1 on a number line, including the point at 1. This visualization matches the interval notation \([1, \infty)\), demonstrating all the valid values for \(x\).
Linear inequalities are applicable in many real-world situations where conditions are set to range limits instead of having exact values.