Problem 17
Question
In Exercises 15 through 26 , find the solution set of the given inequality, and illustrate the solution on the real number line. $$ |3 x-4| \leq 2 $$
Step-by-Step Solution
Verified Answer
\(\frac{2}{3} \leq x \leq 2\)
1Step 1: Understand the Absolute Value Inequality
The inequality \( |3x - 4| \leq 2 \) means that the expression inside the absolute value is between -2 and 2. This can be split into two inequalities: \(-2 \leq 3x - 4 \leq 2\).
2Step 2: Split The Inequality
Rewrite the absolute value inequality as two separate inequalities: \(-2 \leq 3x - 4\) and \(3x - 4 \leq 2\).
3Step 3: Solve the First Inequality
Add 4 to both sides of \((-2 \leq 3x - 4)\) to get \(2 \leq 3x\). Then divide by 3 to get \(\frac{2}{3} \leq x\).
4Step 4: Solve the Second Inequality
Add 4 to both sides of \(3x - 4 \leq 2)\) to get \(3x \leq 6\). Then divide by 3 to get \(x \leq 2\).
5Step 5: Combine the Two Inequalities
Combine the solutions from steps 3 and 4 to get \(\frac{2}{3} \leq x \leq 2\).
6Step 6: Illustrate on the Real Number Line
Draw a number line and shade the region between \(\frac{2}{3}\) and \(2\). Include both endpoints as the original inequality is inclusive.
Key Concepts
Solving InequalitiesReal Number LineAbsolute Value
Solving Inequalities
Inequalities show the relationship between two expressions where one is not necessarily equal to the other.
They can use symbols like <, >, ≤, or ≥.
Solving inequalities is a lot like solving equations, but with some important differences.
When you multiply or divide both sides of an inequality by a negative number, you have to reverse the inequality sign.
Let's look at the example from the exercise.
We start with the inequality \( |3x - 4| \leq 2 \).
The absolute value tells us that \( 3x - 4 \) is between -2 and 2.
This means we are dealing with two connected inequalities: \(-2 \leq3x - 4\leq 2\).
Solving these two separately involves isolating \( x \).
By breaking it down into steps, it’s clear and simple.
They can use symbols like <, >, ≤, or ≥.
Solving inequalities is a lot like solving equations, but with some important differences.
When you multiply or divide both sides of an inequality by a negative number, you have to reverse the inequality sign.
Let's look at the example from the exercise.
We start with the inequality \( |3x - 4| \leq 2 \).
The absolute value tells us that \( 3x - 4 \) is between -2 and 2.
This means we are dealing with two connected inequalities: \(-2 \leq3x - 4\leq 2\).
Solving these two separately involves isolating \( x \).
By breaking it down into steps, it’s clear and simple.
Real Number Line
The real number line is a visual tool that helps us understand the range of values an inequality can take.
It runs infinitely in both directions and represents all real numbers.
When solving inequalities like \( |3x - 4| \leq 2 \), we need to show the solution set on this number line.
First, solve the inequalities to find \( \frac{2}{3} \leq x \leq 2 \).
Using the real number line, you would:
It runs infinitely in both directions and represents all real numbers.
When solving inequalities like \( |3x - 4| \leq 2 \), we need to show the solution set on this number line.
First, solve the inequalities to find \( \frac{2}{3} \leq x \leq 2 \).
Using the real number line, you would:
- Draw a line and mark points at \( \frac{2}{3} \) and \( 2 \).
- Shade the area between these points.
- Include the end points since the inequality is inclusive (≤).
Absolute Value
Absolute value measures the distance of a number from zero, always as a positive value or zero.
It's like asking, 'how far is this number from zero?' without considering direction.
The symbol for absolute value is two vertical lines: \| |\.
For example, \|3x - 4| \ means the distance from zero to \(3x - 4\).
In inequalities, absolute value splits into two cases:
You solve both to find the range of values for \( x \).
Combining these helps us understand where \( x \) lies on the number line and find its solution set.
It's like asking, 'how far is this number from zero?' without considering direction.
The symbol for absolute value is two vertical lines: \| |\.
For example, \|3x - 4| \ means the distance from zero to \(3x - 4\).
In inequalities, absolute value splits into two cases:
- One for the positive distance.
- One for the negative distance.
You solve both to find the range of values for \( x \).
Combining these helps us understand where \( x \) lies on the number line and find its solution set.
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