To find the solution to [tex]\( |3x - 3| < 3 \)[/tex], we need to solve the absolute value inequality. Here’s the step-by-step process:
1. Understand the Absolute Value Inequality:
The inequality [tex]\( |3x - 3| < 3 \)[/tex] means that the expression [tex]\( 3x - 3 \)[/tex] is less than 3 units away from 0. This can be translated into two separate inequalities:
[tex]\[
-3 < 3x - 3 < 3
\][/tex]
2. Break it into Two Inequalities:
We now solve the compound inequality by first isolating [tex]\( 3x - 3 \)[/tex]:
[tex]\[
-3 < 3x - 3 < 3
\][/tex]
3. Add 3 to All Parts of the Inequality:
To isolate [tex]\( 3x \)[/tex], add 3 to all three parts of the inequality:
[tex]\[
-3 + 3 < 3x - 3 + 3 < 3 + 3
\][/tex]
Simplify this to:
[tex]\[
0 < 3x < 6
\][/tex]
4. Divide by 3:
To solve for [tex]\( x \)[/tex], divide all parts of the inequality by 3:
[tex]\[
\frac{0}{3} < \frac{3x}{3} < \frac{6}{3}
\][/tex]
This simplifies to:
[tex]\[
0 < x < 2
\][/tex]
5. Interval Notation:
The solution to the inequality [tex]\( |3x - 3| < 3 \)[/tex] is the set of all [tex]\( x \)[/tex] such that:
[tex]\[
0 < x < 2
\][/tex]
6. Number Line Representation:
On the number line, this solution is represented by an open interval from 0 to 2, not including the endpoints 0 and 2. This means we place open circles at 0 and 2 and shade the region between them.
Based on this solution, you need the number line that has an open circle at 0, an open circle at 2, and the region between them shaded. If you look at the options provided (A, B, C, and D), you should select the one that matches this description.
In conclusion, the number line representing the solution to [tex]\( |3x - 3| < 3 \)[/tex] is the one showing [tex]\( 0 < x < 2 \)[/tex] with open circles at 0 and 2 and shading in between them.
