Problem 51 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|10-4 x|+1 \geq 5$$

Step-by-Step Solution

Answer

x \leq \frac{3}{2} or x \geq \frac{7}{2}

Step 1: Isolate the Absolute Value Expression

Subtract 1 from both sides of the inequality: \[|10-4x| + 1 - 1 \, \geq \, 5 - 1\]Thus, \[|10-4x| \, \geq \, 4\]

Step 2: Set Up Two Inequalities

Recall that if \, \(|A| \, \geq \, B\), then \, \(A \, \geq \, B \, \text{or} \, A \, \leq \, -B\). Therefore, we consider: \[10-4x \, \geq \, 4\] and \[10-4x \, \leq \, -4\]

Step 3: Solve the First Inequality

Solve the inequality \, \(10-4x \, \geq \, 4\): Subtract 10 from both sides: \[10-4x-10 \, \geq \, 4-10\]\[-4x \, \geq \, -6\]Divide both sides by -4 (remember to reverse the inequality symbol): \[\frac{-4x}{-4} \, \leq \, \frac{-6}{-4}\]\[x \, \leq \, \frac{3}{2}\]

Step 4: Solve the Second Inequality

Solve the inequality \, \(10-4x \, \leq \, -4\): Subtract 10 from both sides: \[10-4x-10 \, \leq \, -4-10\]\[-4x \, \leq \, -14\]Divide both sides by -4 (remember to reverse the inequality symbol): \[\frac{-4x}{-4} \, \geq \, \frac{-14}{-4}\]\[x \, \geq \, \frac{7}{2}\]

Step 5: Write the Final Solution

Combine the solutions from Step 3 and Step 4: \[x \, \leq \, \frac{3}{2} \, \text{or} \, x \, \geq \, \frac{7}{2}\]Thus, \[x \, \in \, (-\infty, \frac{3}{2}] \, \cup \, [\frac{7}{2}, \infty)\]

Key Concepts

Isolate the Absolute Value Expression

To start solving the inequality involving an absolute value, we need to first isolate the absolute value part of the equation. This step simplifies the process and sets the stage for solving the inequality properly.

Given the inequality \(|10 - 4x| + 1 \geq 5\), we see an extra constant on the left side of the inequality. We need to get rid of it by performing basic arithmetic operations, in this case, subtraction:

Subtract 1 from both sides:
\[|10 - 4x| + 1 - 1 \geq 5 - 1\]

This simplifies to:
\[|10 - 4x| \geq 4\]

Now, the absolute value expression is isolated on one side of the inequality.

Set Up Two Inequalities

Absolute value inequalities can be broken down into two separate inequalities. This method leverages the fact that the absolute value definition reflects two possible scenarios.

For the expression \(|A| \geq B\), it implies that either
\[A \geq B \text{ or } A \leq -B\]

Applying this to our isolated absolute value \(|10 - 4x| \geq 4\), we get:

• \[10 - 4x \geq 4\]
• \[10 - 4x \leq -4\]

Now, we have two separate inequalities to work with.

Solve the Inequalities

With the two inequalities set up, the next step is to solve each inequality individually. Let's break it down one by one.

First Inequality:
\[10 - 4x \geq 4\]

1. Subtract 10 from both sides:
\[10 - 4x - 10 \geq 4 - 10\]
This simplifies to:
\[-4x \geq -6\]
2. Divide both sides by -4 (remember to reverse the inequality symbol):
\[\frac{-4x}{-4} \leq \frac{-6}{-4}\]
This results in:
\[x \leq \frac{3}{2}\]

Second Inequality:
\[10 - 4x \leq -4\]

1. Subtract 10 from both sides:
\[10 - 4x - 10 \leq -4 - 10\]
This simplifies to:
\[-4x \leq -14\]
2. Divide both sides by -4 (remember to reverse the inequality symbol):
\[\frac{-4x}{-4} \geq \frac{-14}{-4}\]
This results in:
\[x \geq \frac{7}{2}\]

Combined Solution

Once we've solved both individual inequalities, we need to combine the solutions.

The two solutions we obtained were:
1. \([x \leq \frac{3}{2}]\)
2. \([x \geq \frac{7}{2}]\)

These solutions suggest two distinct ranges for x. Thus, combining them, we write:
\[x \in (-\infty, \frac{3}{2}] \cup [\frac{7}{2}, \infty)\]

This shows the values of x that satisfy the original absolute value inequality.

If you visualize the number line:

• For values less than or equal to \frac{3}{2}, the expression holds true.
• For values greater than or equal to \frac{7}{2}, the expression also holds true.

Thus, the combined solution provides a comprehensive set of values satisfying the initial condition.