Problem 37 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 4(3 x-2) \leq-2(x+3)+12 $$

Step-by-Step Solution

Answer

The solution is \(x \leq 1\), or \((-\infty, 1]\).

Step 1: Distribute Constants

Expand each side of the inequality by distributing the constants through the parentheses:Left side: \(4(3x-2) = 12x - 8\)Right side: \(-2(x + 3) + 12 = -2x - 6 + 12\)

Step 2: Simplify Both Sides

Combine like terms on the right side:Right side simplifies to: \(-2x - 6 + 12 = -2x + 6\)

Step 3: Rearrange Terms

Keep the inequality sign the same while rearranging terms:\(12x - 8 \leq -2x + 6\)

Step 4: Add/Subtract Variables

Add \(2x\) to both sides to gather the \(x\)-terms on one side:\(12x + 2x - 8 \leq 6\)Which simplifies to: \(14x - 8 \leq 6\)

Step 5: Isolate the Variable

Add \(8\) to both sides to isolate the \(x\) term:\(14x - 8 + 8 \leq 6 + 8\)Which becomes: \(14x \leq 14\)

Step 6: Solve for x

Divide each side by \(14\) to solve for \(x\):\(x \leq \frac{14}{14}\)Thus, \(x \leq 1\).

Step 7: Solution in Interval Notation

Express the solution \(x \leq 1\) in interval notation:The solution set is \((-\infty, 1]\).

Step 8: Graph the Solution Set

Draw a number line. Shade the line to the left of \(1\) and include a closed dot on \(1\) to indicate that \(1\) is part of the solution set.

Key Concepts

Solution Set

In mathematics, finding the solution set of an inequality involves identifying all possible values that satisfy the condition laid out by the inequality equation. When we solve the inequality \(4(3x-2) \leq -2(x+3) + 12\), we follow a series of steps to determine the values of \(x\).

• Initially, we work by expanding and simplifying both sides of the inequality to isolate the variable.
• Once simplified, the solution is determined, which in this case means calculating that \(x \leq 1\).

This means that any number less than or equal to 1 will make the original inequality true.
A solution set encompasses all these valid values for \(x\), allowing you to understand the range of solutions in context.

Interval Notation

Interval notation provides a concise way to represent sets of numbers, particularly useful here when dealing with inequalities. In our solution \(x \leq 1\), interval notation becomes a handy tool.
Interval notation is written with parentheses \(()\) and brackets \( [] \) to show the bounds of the solution set:

• Parentheses \(()\) indicate that a number is not part of a set
• Brackets \([]\) indicate that a number is included in a set

For the inequality \(x \leq 1\), the interval notation is \((-fty, 1]\). Here, \(1\) is included, shown by a bracket \( ]\), and \(-\infty\) suggests that the values stretch infinitely towards negative numbers, never quite reaching infinity, thus shown with a parenthesis. Understanding interval notation helps in visualizing and communicating the bounds of a solution set efficiently.

Graphing Inequalities

Graphing inequalities allows you to see the solution set visually. In our example, \(x \leq 1\), graphing helps illustrate this range effectively.
To graph the solution,

• Start by drawing a number line
• Locate the number \(1\) on this line
• Since \(x\) can equal \(1\) (indicating \(1\) is included in the solution set), place a filled or closed dot at \(1\)
• Shade the line to the left of \(1\) to illustrate all numbers that are less than \(1\)

This simple graph displays all possible solutions quickly and clearly. Graphs serve as a practical tool for understanding and checking the validity of a solution set derived from inequalities.