91Ó°ÊÓ

A number line is a visual tool used in mathematics to represent numbers in a straight, linear format. Imagine a horizontal line that extends infinitely in both directions. Key characteristics include numbers ordered sequentially with zero typically in the center.
Negative numbers are placed on the left side of zero, while positive numbers are on the right. This setup helps us easily understand and graph the relationships between numbers.

• Each point on the number line corresponds to a unique real number.
• The spacing between points helps illustrate the magnitude or size of numbers.

To graph, such as in the exercise given, we use dots, circles, and arrows to visually indicate solutions. For example, in the solution to the inequality \( x \geq 9 \), we place a closed circle at 9 to signify that 9 is included in the solution set, and an arrow pointing right to show all numbers greater than 9 are also part of this solution.

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It allows us to create equations and inequalities to solve real-world problems.
In algebra, letters are used to represent numbers, often called unknowns. These letters, or variables, can be manipulated using mathematical operations to solve problems.

• Identification of terms is important. Each symbol or number separated by operators like \(+\) or \(-\) is a term, such as \(3x\) or \(7\).
• Rearranging terms involves moving terms from one side of an equation to the other, adhering to basic arithmetic principles to maintain balance.
• Simplifying expressions means condensing terms and eliminating redundancy by performing operations.

The original exercise demonstrates these ideas. By moving terms around and combining like terms, step by step, the inequality becomes simpler and more solvable.

Solving inequalities involves finding the value of a variable that makes the inequality true. This is similar to solving equations, but with extra rules for flipping inequality signs when multiplying or dividing by a negative number.
In inequalities like \(3x + 7 \leq 4x - 2\), our goal is to isolate the variable on one side of the inequality to discover the range of values it can take.
The process includes:

• Collecting like terms on one side, as done in Step 1 where \(3x\) was subtracted from both sides to collect \(x\) terms together.
• Solving for the variable by isolating it, which involves removing constants, such as adding 2 to both sides in Step 2.
• Interpreting the solution appropriately. This results in an inequality like \(x \geq 9\), which translates to both a numerical range and a manner of representing this range graphically.

Once inequalities are solved, their solutions are often visualized on a number line, aiding understanding of the set of all possible solutions.