Problem 22 - The Calculus with Analytic Geometry

Question

In Exercises 15 through 26 , find the solution set of the given inequality, and illustrate the solution on the real number line. $$ |3 x|>|6-3 x| $$

Step-by-Step Solution

Answer

x < 1 or x > 1

Step 1 - Set Up Inequality

Start by setting up the inequality |3x| > |6 - 3x|

Step 2 - Analyze Absolute Values

Consider the expression inside the absolute value functions separately for different cases. The first case to consider is when the expressions inside the absolute values are both positive or zero, both negative, or one positive and one negative.

Step 3 - Case 1: Both Positive

For the case when both sides are positive or zero: 3x > 6 - 3x Solve for x: 3x + 3x > 6 6x > 6 x > 1

Step 4 - Case 2: Both Negative

For the case when expressions inside the absolute values are negative (or zero): -3x > -(6 - 3x) Simplifying this: -3x > -6 + 3x -3x - 3x > -6 -6x > -6 x < 1

Step 5 - Combine Results

Combine the results from both cases. Check for intervals that satisfy both conditions. The solution is the union of both results. However, isolate where inconsistencies occur: x ≠ 1.

Step 6 - Illustrate Solution

Draw a number line and mark the critical point, which is x = 1. The solution to the inequality is shown by shading the regions where x < 1 and where x > 1, but not including x = 1.

Key Concepts

absolute values

Absolute values deal with the 'distance' of a number from zero on the number line. They are always non-negative.
When you see an absolute value symbol, for example, \(|a|\), this means that you want the positive version of whatever is inside it.
For example:
\(|3| = 3\) because 3 is already positive.
\(|-4| = 4\) since we want the positive distance, so we drop the negative sign.
The expression \(|3x|\) literally means 'the absolute value of 3 times x'. This ensures that even if 3x is negative, its absolute value makes it positive.

inequality solution

Solving absolute value inequalities means finding all x-values that make the inequality true.
There can be different cases to consider:

• Both expressions inside the absolute values are non-negative.
• Both expressions inside the absolute values are non-positive.
• One expression inside the absolute value is positive, the other one is negative.

Here's how we handle each case for the given problem \( |3x| > |6-3x| \):

Case 1: Both positive
\ |3x| = 3x \ and \ |6-3x| = 6-3x \
So, 3x > 6 - 3x
Adding 3x to both sides, we get: 6x > 6
Dividing by 6, we find: x > 1

Case 2: Both negative
\ |3x| = -3x \ and \ |6-3x| = -(6-3x) = -6 + 3x \
Simplifying: -3x > -6 + 3x
Subtracting 3x from both sides: -6x > -6
Dividing by -6 (remember to reverse the inequality when dividing by a negative number), we get: x < 1

Combine Results:
Combining both cases: x > 1 or x < 1
This means every x except x=1.

real number line

The real number line is a way to visually represent all real numbers.
Each point on the line corresponds to a real number, and it's typically arranged in increasing order from left to right.
To illustrate the solution of an inequality on the real number line:

• Identify key points (in this case, x=1).
• Determine which intervals satisfy the inequality.
• Shade or mark those regions.

For the inequality \(|3x|>|6-3x|\), the solution was x<1 or x>1. On the number line:

• The region left of x=1 (not including 1) is shaded.
• The region right of x=1 (not including 1) is also shaded.
• x=1 is NOT included in the solution, so we mark it with an open circle.