To solve the compound inequality [tex]\(3x - 5 > 13x - 5 > 1\)[/tex], we need to break it down into two separate inequalities and solve them individually.
### Step 1: Split the compound inequality
The compound inequality [tex]\(3x - 5 > 13x - 5 > 1\)[/tex] can be expressed as two inequalities:
1. [tex]\(3x - 5 > 13x - 5\)[/tex]
2. [tex]\(13x - 5 > 1\)[/tex]
### Step 2: Solve the first inequality [tex]\(3x - 5 > 13x - 5\)[/tex]
To isolate [tex]\(x\)[/tex], we follow these steps:
1. Subtract [tex]\(-5\)[/tex] from both sides:
[tex]\[3x - 5 + 5 > 13x - 5 + 5\][/tex]
[tex]\[3x > 13x\][/tex]
2. Subtract [tex]\(13x\)[/tex] from both sides:
[tex]\[3x - 13x > 0\][/tex]
[tex]\[-10x > 0\][/tex]
3. Divide both sides by [tex]\(-10\)[/tex]. Remember, when dividing by a negative number, the inequality sign flips.
[tex]\[x < 0\][/tex]
So, the solution to [tex]\(3x - 5 > 13x - 5\)[/tex] is:
[tex]\[x < 0\][/tex]
### Step 3: Solve the second inequality [tex]\(13x - 5 > 1\)[/tex]
To isolate [tex]\(x\)[/tex], we follow these steps:
1. Add [tex]\(5\)[/tex] to both sides:
[tex]\[13x - 5 + 5 > 1 + 5\][/tex]
[tex]\[13x > 6\][/tex]
2. Divide both sides by [tex]\(13\)[/tex]:
[tex]\[x > \frac{6}{13}\][/tex]
So, the solution to [tex]\(13x - 5 > 1\)[/tex] is:
[tex]\[x > \frac{6}{13}\][/tex]
### Step 4: Combine the solutions
We need the solutions to both inequalities to be satisfied simultaneously. The solutions are:
1. [tex]\(x < 0\)[/tex]
2. [tex]\(x > \frac{6}{13}\)[/tex]
Since there are no overlapping values that satisfy both inequalities simultaneously, this compound inequality has two different ranges but no common range.
Therefore, the final combined solution to the inequality [tex]\(3x - 5 > 13x - 5 > 1\)[/tex] is:
[tex]\[x < 0 \text{ or } x > \frac{6}{13}\][/tex]
The solution set can be written as:
[tex]\[(-\infty, 0) \cup \left(\frac{6}{13}, \infty\right)\][/tex]
