Problem 33 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 5-3(2 x-6) \geq-1 $$

Step-by-Step Solution

Answer

The solution is \(x \leq 4\), interval notation is \((-\infty, 4]\).

Step 1: Simplify the Expression

Start by expanding the equation. The original inequality is \(5 - 3(2x - 6) \geq -1\). Distribute the \(-3\) across the parentheses, which gives you \(5 - 6x + 18 \geq -1\).

Step 2: Combine Like Terms

Combine the constant terms in the inequality. Combine \(5 + 18\) to get \(23\), resulting in the new inequality \(23 - 6x \geq -1\).

Step 3: Isolate the Variable Term

Subtract \(23\) from both sides to isolate the terms involving \(x\). This gives you \(-6x \geq -24\).

Step 4: Solve for the Variable

Divide each side of the inequality by \(-6\). Remember that dividing by a negative number reverses the inequality sign: \(x \leq 4\).

Step 5: Graph the Solution Set

On a number line, draw a closed circle at \(x = 4\) and shade the number line to the left of 4, which represents all numbers \(x\) that are less than or equal to 4.

Step 6: Write in Interval Notation

The solution set in interval notation is \(( -\infty, 4 ]\), indicating all numbers less than or equal to 4.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication) that stand for values. In the inequality given, the expression is \(5 - 3(2x - 6)\). Here, you have several components:

• The number 5, which is a constant.
• The variable \(x\), whose value we need to find.
• An expression in parentheses, \(2x - 6\), which you multiply by \(-3\).

The process of solving such an expression involves expanding and simplifying it to make it easier to work with. Expansion, as shown in the solution, means removing the parentheses by distributing the multiplication over the terms within. Simplification involves combining like terms—terms that contain the same variable raised to the same power or constants—to form a more concise expression. This forms the basis of solving inequalities and equations in algebra.

Interval Notation Explained

Interval notation is a succinct way of representing sets of numbers, especially solutions to inequalities. In this notation:

• Parentheses \(( )\) denote that an endpoint is not included in the interval.
• Brackets \([ ]\) indicate that an endpoint is included.
• The symbol \(-\infty\) or \(+\infty\) is used to indicate that the set extends indefinitely in a negative or positive direction.

For the solved inequality \(x \leq 4\), the solution set is expressed as \((-\infty, 4]\). This means that the set includes all real numbers less than or equal to 4. The bracket at 4 shows that 4 itself is included in the solution set, whereas the parenthesis near \(-\infty\) indicates that there is no lower bound included in the set. This concise representation is very useful for indicating ranges of values in mathematical solutions.

Graphing Inequalities

Graphing inequalities helps visualize the range of solutions on a number line. Here's how you graph an inequality such as \(x \leq 4\):

• Draw a number line with appropriate markings including the number 4.
• Place a closed circle at 4, as the inequality \(\leq\) indicates that 4 is part of the solution set.
• Shade the number line to the left of 4, which captures all values that satisfy \(x \leq 4\).

This graphical representation serves as a clear and intuitive method to visualize how values progress relative to each other. By graphing inequalities, you turn abstract algebraic solutions into something you can see, making it easier to understand and interpret the possible values for \(x\). Graphs help solve inequalities by providing a method to quickly verify intervals, especially useful in more complex equations.