Problem 67 - Algebra for College Students

Question

Solve each equation or inequality. Graph the solution set. $$ |10-12 x| \geq 4 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, \frac{1}{2}] \cup [\frac{7}{6}, \infty)\).

Step 1 - Understand the Absolute Value Inequality

The given inequality is \(|10 - 12x| \geq 4\). This means that the expression inside the absolute value can be either greater than or equal to 4 or less than or equal to -4.

Step 2 - Split the Inequality

Break down the absolute value inequality into two separate inequalities:1. \(10 - 12x \geq 4\)2. \(10 - 12x \leq -4\)

Step 3 - Solve the First Inequality

Solve \(10 - 12x \geq 4\):First, subtract 10 from both sides:\(10 - 12x - 10 \geq 4 - 10\)This simplifies to:\(-12x \geq -6\)Then, divide both sides by -12 and reverse the inequality sign:\(x \leq \frac{1}{2}\)

Step 4 - Solve the Second Inequality

Solve \(10 - 12x \leq -4\):First, subtract 10 from both sides:\(10 - 12x - 10 \leq -4 - 10\)This simplifies to:\(-12x \leq -14\)Then, divide both sides by -12 and reverse the inequality sign:\(x \geq \frac{7}{6}\)

Step 5 - Combine the Solutions

Combine the two solutions: \(x \leq \frac{1}{2}\) and \(x \geq \frac{7}{6}\). Since the solutions do not overlap, the solution set is the union of the two intervals: \((-\infty, \frac{1}{2}] \cup [\frac{7}{6}, \infty)\).

Step 6 - Graph the Solution Set

Graph the solution set on a number line. Indicate the intervals \((-\infty, \frac{1}{2}]\) and \([\frac{7}{6}, \infty)\) with filled circles at \(x = \frac{1}{2}\) and \(x = \frac{7}{6}\). Shade the regions extending to negative infinity from \(\frac{1}{2}\) and to positive infinity from \(\frac{7}{6}\).

Key Concepts

solving inequalities

Solving absolute value inequalities involves understanding that the absolute value of an expression can be interpreted in two ways. For the inequality \(|10 - 12x| \geq 4\), this means we need to consider two separate conditions.

An absolute value inequality \[ |A| \geq B \] can be split into:

• \( A \geq B \)
• \( A \leq -B \)

Applying this principle, we get:

• \( 10 - 12x \geq 4 \)
• \( 10 - 12x \leq -4 \)

After simplifying these inequalities, we get two individual inequalities to solve. Always make sure to reverse the inequality sign when you divide by a negative number.

graphing solution sets

Once you solve the inequalities, graphing the solution set correctly is crucial. For the given problem, the combined solutions to the inequalities are \[ x \leq \frac{1}{2} \] and \[ x \geq \frac{7}{6} \].

To graph these, follow these steps:

• Draw a number line.
• Mark the points \(\frac{1}{2}\) and \(\frac{7}{6}\) on the number line.
• Use filled circles at \(\frac{1}{2}\) and \(\frac{7}{6}\) to indicate that these values are included.
• Shade the regions to the left of \(\frac{1}{2}\) and to the right of \(\frac{7}{6}\).

The final graph will show the intervals extending from \(-\infty \) to \(\frac{1}{2} \) and from \( \frac{7}{6} \) to \( \infty \).

algebraic expressions

Understanding how to manipulate algebraic expressions is essential in solving absolute value inequalities. Start by isolating the absolute value term on one side of the inequality.

For \(|10 - 12x| \geq 4\), isolate \(10 - 12x\). This gives two separate inequalities to solve. First, handle \(10 - 12x \geq 4\):

• Subtract 10 from both sides to get \(-12x \geq -6\).
• Divide by \(-12\), remembering to reverse the inequality sign: \(\frac{x}{-\frac{12}{12}} \leq \frac{-6}{-12}\), simplifying to \( x \leq \frac{1}{2}\).

Second, solve \(10 - 12x \leq -4\):

• Subtract 10 from both sides to get \(-12x \leq -14\).
• Divide by \(-12\), reversing the inequality sign: \(\frac{x}{\frac{-12}{12}} \geq \frac{-14}{-12}\), simplifying to \( x \geq \frac{7}{6}\).

Combining these solutions gives the complete answer.