Answer :
To solve the inequality \( |1 - 6x| - 6 < -3 \), we need to simplify it step by step.
### Step 1: Isolate the Absolute Value
We start by isolating the absolute value expression:
[tex]\[ |1 - 6x| - 6 < -3 \][/tex]
Add 6 to both sides to isolate the absolute value:
[tex]\[ |1 - 6x| < 3 \][/tex]
### Step 2: Understand the Absolute Value Inequality
The inequality \( |1 - 6x| < 3 \) means that the expression inside the absolute value, \(1 - 6x\), is within 3 units of 0. This can be expressed as two separate inequalities:
[tex]\[ -3 < 1 - 6x < 3 \][/tex]
### Step 3: Break it into Two Inequalities
Now, we break this compound inequality into two separate inequalities:
1. \( 1 - 6x < 3 \)
2. \( 1 - 6x > -3 \)
### Step 4: Solve Each Inequality Individually
#### Inequality 1: \( 1 - 6x < 3 \)
Subtract 1 from both sides:
[tex]\[ -6x < 2 \][/tex]
Divide by -6 (and reverse the inequality sign):
[tex]\[ x > -\frac{1}{3} \][/tex]
#### Inequality 2: \( 1 - 6x > -3 \)
Subtract 1 from both sides:
[tex]\[ -6x > -4 \][/tex]
Divide by -6 (and reverse the inequality sign):
[tex]\[ x < \frac{2}{3} \][/tex]
### Step 5: Combine the Solutions
Combining the results from both inequalities, we get:
[tex]\[ -\frac{1}{3} < x < \frac{2}{3} \][/tex]
### Graph the Solution Set
To graph the solution set, we focus on the interval \(-\frac{1}{3} < x < \frac{2}{3}\):
- Draw a number line.
- Mark the points \(-\frac{1}{3}\) and \(\frac{2}{3}\) on the number line.
- Since the inequality does not include the endpoints, use open circles at \(-\frac{1}{3}\) and \(\frac{2}{3}\).
- Shade the region between \(-\frac{1}{3}\) and \(\frac{2}{3}\).
In summary, the solution to the inequality \( |1 - 6x| - 6 < -3 \) is:
[tex]\[ -\frac{1}{3} < x < \frac{2}{3} \][/tex]
And the graph of the solution set is an open interval between \(-\frac{1}{3}\) and \(\frac{2}{3}\) on the number line:
[tex]\[ \begin{array}{c} \text{ } \\ \underset{-1/3 \quad 0 \quad 2/3}{\overset{\ \ \ \ \ \small \bigcirc \quad \quad \small \bigcirc}{\line(1,0){150}}} \end{array} \][/tex]
### Step 1: Isolate the Absolute Value
We start by isolating the absolute value expression:
[tex]\[ |1 - 6x| - 6 < -3 \][/tex]
Add 6 to both sides to isolate the absolute value:
[tex]\[ |1 - 6x| < 3 \][/tex]
### Step 2: Understand the Absolute Value Inequality
The inequality \( |1 - 6x| < 3 \) means that the expression inside the absolute value, \(1 - 6x\), is within 3 units of 0. This can be expressed as two separate inequalities:
[tex]\[ -3 < 1 - 6x < 3 \][/tex]
### Step 3: Break it into Two Inequalities
Now, we break this compound inequality into two separate inequalities:
1. \( 1 - 6x < 3 \)
2. \( 1 - 6x > -3 \)
### Step 4: Solve Each Inequality Individually
#### Inequality 1: \( 1 - 6x < 3 \)
Subtract 1 from both sides:
[tex]\[ -6x < 2 \][/tex]
Divide by -6 (and reverse the inequality sign):
[tex]\[ x > -\frac{1}{3} \][/tex]
#### Inequality 2: \( 1 - 6x > -3 \)
Subtract 1 from both sides:
[tex]\[ -6x > -4 \][/tex]
Divide by -6 (and reverse the inequality sign):
[tex]\[ x < \frac{2}{3} \][/tex]
### Step 5: Combine the Solutions
Combining the results from both inequalities, we get:
[tex]\[ -\frac{1}{3} < x < \frac{2}{3} \][/tex]
### Graph the Solution Set
To graph the solution set, we focus on the interval \(-\frac{1}{3} < x < \frac{2}{3}\):
- Draw a number line.
- Mark the points \(-\frac{1}{3}\) and \(\frac{2}{3}\) on the number line.
- Since the inequality does not include the endpoints, use open circles at \(-\frac{1}{3}\) and \(\frac{2}{3}\).
- Shade the region between \(-\frac{1}{3}\) and \(\frac{2}{3}\).
In summary, the solution to the inequality \( |1 - 6x| - 6 < -3 \) is:
[tex]\[ -\frac{1}{3} < x < \frac{2}{3} \][/tex]
And the graph of the solution set is an open interval between \(-\frac{1}{3}\) and \(\frac{2}{3}\) on the number line:
[tex]\[ \begin{array}{c} \text{ } \\ \underset{-1/3 \quad 0 \quad 2/3}{\overset{\ \ \ \ \ \small \bigcirc \quad \quad \small \bigcirc}{\line(1,0){150}}} \end{array} \][/tex]
