Problem 69 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -32 \leq 14(12 x-1)+34<32 $$

Step-by-Step Solution

Answer

\( \left[ \frac{-13}{42}, \frac{1}{14} \right) \)

Step 1: Expand and Simplify the Inequality

First, distribute the 14 in the expression inside the inequality. Break it down into two separate inequalities:1. \[-32 \leq 14(12x-1) + 34\]Distribute the 14:\[-32 \leq 168x - 14 + 34\] Combine like terms:\[-32 \leq 168x + 20\]2. \[168x - 14 + 34 < 32\]Combine like terms:\[168x + 20 < 32\]

Step 2: Solve the First Inequality

Now, solve the first inequality from Step 1:\[-32 \leq 168x + 20\] Subtract 20 from both sides:\[-52 \leq 168x\] Divide by 168:\[x \geq \frac{-52}{168}\]Simplify the fraction:\[x \geq \frac{-13}{42}\]

Step 3: Solve the Second Inequality

Next, solve the second inequality from Step 1:\[168x + 20 < 32\] Subtract 20 from both sides:\[168x < 12\] Divide by 168:\[x < \frac{12}{168}\]Simplify the fraction:\[x < \frac{1}{14}\]

Step 4: Combine Solution for the Inequality

Combine the solutions from Step 2 and Step 3 to find the range of values for \(x\):\[ \frac{-13}{42} \leq x < \frac{1}{14} \]This represents the solution set for the inequality.

Step 5: Graph the Solution Set

Draw a number line and shade the region that matches \( \frac{-13}{42} \leq x < \frac{1}{14} \). Notice that \(\frac{-13}{42}\) has a closed dot indicating it is included, while \(\frac{1}{14}\) has an open dot indicating it is not included.

Step 6: Write the Solution in Interval Notation

Express the solution set in interval notation:\[ \left[ \frac{-13}{42}, \frac{1}{14} \right) \]This indicates the interval includes \(\frac{-13}{42}\) but does not include \(\frac{1}{14}\).

Key Concepts

Interval Notation

Interval notation is a way to describe the set of solutions of inequalities using interval brackets. It simplifies the expression of ranges where a variable falls between two numbers. Consider the expression \( \frac{-13}{42} \leq x < \frac{1}{14} \). In interval notation, this can be expressed as \( \left[ \frac{-13}{42}, \frac{1}{14} \right) \).
Here’s a brief breakdown of the different types of brackets used in interval notation:

• [ ]: These brackets indicate that the endpoint is included in the interval. For example, \( [a, b] \) means "\( a \)" and "\( b \)" are part of the solution set.
• ( ): These brackets show that the endpoint is not included. Thus, \( (a, b) \) means neither "\( a \)" nor "\( b \)" is included.

In this specific solution, \( \left[ \frac{-13}{42}, \frac{1}{14} \right) \) means that \( \frac{-13}{42} \) is part of the solution, while \( \frac{1}{14} \) is not. This clear notation efficiently conveys which values belong to the solution set.

Graphical Representation

Visualizing inequalities with a graph provides a straightforward way to understand solution sets. When an inequality is solved, a number line is often used to represent the range of possible values. Here’s how to graph the solution set \( \frac{-13}{42} \leq x < \frac{1}{14} \):

• Draw a number line where both \( \frac{-13}{42} \) and \( \frac{1}{14} \) are marked as key points.
• Place a closed dot on \( \frac{-13}{42} \). This indicates that \( x \) can equal \( \frac{-13}{42} \) since the inequality is \( \leq \).
• Place an open dot on \( \frac{1}{14} \). This shows that \( x \) cannot equal \( \frac{1}{14} \), as it is just "less than".

Shade the region between these two points to show all the possible values that \( x \) can take.Using a graphical representation makes it easier for you to visually verify the solutions and comprehend the scope of the interval in question.

Solution Set

A solution set contains all the possible solutions that satisfy a given inequality or equation. For the problem \(-32 \leq 14(12 x-1)+34 <32 \), the solution set comprises values of \( x \) that make this condition true.
After simplifying and solving the inequalities step by step, we found that:

• \( x \geq \frac{-13}{42} \) from the first inequality.
• \( x < \frac{1}{14} \) from the second inequality.

Combining these results gives us the final solution set: \( \frac{-13}{42} \leq x < \frac{1}{14} \).
This means any value of \( x \) within this range satisfies the original inequality. Understanding the concept of a solution set helps in solving not just single inequalities but also systems of inequalities, offering a way to evaluate multiple constraints simultaneously.