Problem 40 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 3(x-2)+5>2(3 x+5)+2 $$

Step-by-Step Solution

Answer

\(x < -\frac{13}{3}\); Interval: \((-\infty, -\frac{13}{3})\)

Step 1: Distribute and Simplify

Start by distributing the numbers outside the parentheses on both sides of the inequality: \[ 3(x-2) = 3x - 6 \]\[ 2(3x+5) = 6x + 10 \]Now substitute these into the inequality:\[ 3x - 6 + 5 > 6x + 10 + 2 \]Next, simplify both sides:\[ 3x - 1 > 6x + 12 \]

Step 2: Isolate the Variable

To solve for \(x\), start by moving all terms involving \(x\) to one side of the inequality and constant terms to the other side. Subtract \(6x\) from both sides:\[ 3x - 6x - 1 > 12 \]This simplifies to:\[ -3x - 1 > 12 \]

Step 3: Solve for x

Now, isolate \(x\) by adding 1 to both sides:\[ -3x > 13 \]Then divide by -3, remembering to flip the inequality sign:\[ x < -\frac{13}{3} \]

Step 4: Graph the Solution

Draw a number line and mark the point \(-\frac{13}{3}\). Since the inequality is \(x < -\frac{13}{3}\), draw an open circle at \(-\frac{13}{3}\) and shade to the left, indicating all numbers less than \(-\frac{13}{3}\) are solutions.

Step 5: Write in Interval Notation

Express the solution set in interval notation. Since \(x\) is less than \(-\frac{13}{3}\), the interval is:\[ (-\infty, -\frac{13}{3}) \]

Key Concepts

Graphing Inequalities

When it comes to graphing inequalities, we use a number line to visually represent the solution. This helps us easily see which values satisfy the inequality condition. In our exercise, the inequality is solved to find that \( x < -\frac{13}{3} \). Here’s how we graph that:

• Start by drawing a horizontal number line.
• Determine the key point from the inequality, which is \(-\frac{13}{3}\) in this case.
• Place an open circle at this point to indicate that \(-\frac{13}{3}\) is not included in the solution set, as the inequality is strict (\(< \) rather than \( \leq \)).
• Shade all the area to the left of this point. This shading shows that all values that are less than \(-\frac{13}{3}\) are part of the solution set.

Graphing inequalities can help you visualize which parts of the number line are included as solutions. It makes it easier to determine the range of possible solutions and provides a clear, concise depiction of inequality solutions.

Interval Notation

Interval notation is a shorthand way to express solutions for inequalities, and it's often used alongside graphing. Using this method, we succinctly describe the set of numbers that solve an inequality. For our current inequality solution, \( x < -\frac{13}{3} \), we can express the solution set in interval notation as:

• The left bound is \(-\infty\), indicating that values extend endlessly in the negative direction.
• The right bound is \(-\frac{13}{3}\), the point where the inequality transitions from true to false.
• Use a parenthesis \((\) at \(-\frac{13}{3}\) to show it is not included in the solution set.

Therefore, the interval notation for this inequality is \((-\infty, -\frac{13}{3})\). This notation is clean and efficient, providing a clear way to express ranges without needing a long explanation. It is especially useful for communicating solutions in mathematical contexts succinctly.

Algebraic Expressions

Algebraic expressions form the backbone of algebra and involve variables, numbers, and operations. They are crucial in solving inequalities. For instance, the original exercise begins with a complex expression:

• Apply the distributive property: This step involves distributing the coefficients into the expressions within parentheses, i.e., \(3(x-2)\) becomes \(3x - 6\), and \(2(3x+5)\) becomes \(6x + 10\).
• Simplify the expressions: Combine like terms to reduce the complexity of the inequality to \(3x - 1 > 6x + 12\).
• Manipulate the expression: Isolate the variable on one side by operations such as subtraction and division. For instance, subtracting \(6x\) results in \(-3x - 1 > 12\).

Understanding how to work with algebraic expressions is essential to solving any inequality. Each operation should be conducted systematically, ensuring the integrity of the initial problem while transforming it to find the solution.