91Ó°ÊÓ

Graphing inequalities is a fantastic way to visually represent the solution to an inequality on a number line. To graph an inequality, first determine if the inequality involves a 'less than' (<) or 'greater than' (>) sign, or if it includes 'less than or equal to' (≤) or 'greater than or equal to' (≥). Each of these symbols affects how you graph the range of solutions.

• If the inequality uses 'less than' (<) or 'greater than' (>), you represent this on the number line with an open circle, indicating the endpoint is not included in the solution.
• If it involves 'less than or equal to' (≤) or 'greater than or equal to' (≥), use a closed circle to include the endpoint in the solution set.

For instance, in the exercise \(-8 > x\), the inequality suggests that x must be less than -8. An open circle is drawn at -8, and a line extends to the left, highlighting that all numbers to the left of -8 are solutions to this inequality. This makes it clear where the solution set lies on the number line.

Algebraic expressions are mathematical phrases that include numbers, operations, and variables. They play a critical role in solving inequalities. Understanding how to manipulate these expressions is vital for finding the solution set of an inequality.

• Variables (like x) are symbols representing an unknown quantity. Solving inequalities often involves isolating this variable.
• Terms are parts of the expression separated by addition or subtraction. In \(2x - 6 > 4x + 10\), \(2x\) and \(-6\) are terms on the left, while \(4x\) and \(+10\) are terms on the right.
• Coefficients are numbers in front of the variables, such as 2 in \(2x\).
• Constants are numbers without variables, such as \(-6\) and \(+10\) in the inequality.

When solving the exercise, understanding how to manipulate these expressions helped in transferring terms across the inequality and simplifying to find the solution for x.

The distributive property is a fundamental algebraic tool used to multiply a single term across terms inside parentheses. It states that multiplying a sum by a number gives the same result as multiplying each addend separately by the number and then adding the products.For example, given \(2(x - 3)\), the distributive property allows you to multiply \(2\) by \(x\) and \(2\) by \(-3\), resulting in \(2x - 6\). This is a crucial step in simplifying and solving algebraic expressions and inequalities.

• It helps to break down complex expressions, making it easier to solve or simplify them.
• It is particularly useful in algebra when you need to not only simplify expressions but also solve equations and inequalities.

In the original exercise, using the distributive property was the first step to transform \(2(x-3)\) into \(2x - 6\). This simplification was essential for proceeding with solving the inequality \(2x-6 > 4x + 10\). Understanding and applying the distributive property correctly is vital to ensure accuracy throughout your calculations.