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Part 1
Solve the inequality, and graph the solution on a number line.
3x-3<3(x-1)
To solve the inequality 3x-3 < 3(x-1), we will first simplify the expression on both sides and then solve for x.
First, distribute 3 on the right side:
3x - 3 < 3x - 3
Next, let's isolate the variable x by subtracting 3x from both sides:
-3 < -3
This inequality -3 < -3 is not true since -3 is not less than -3. Therefore, there is no solution for this inequality.
If we were to graph this inequality on the number line, we would not shade any region because there is no solution.
To solve the inequality 3x - 3 < 3(x - 1), we will start by simplifying both sides of the inequality.
Step 1: Distribute the 3 on the right side of the inequality:
3x - 3 < 3x - 3
Step 2: Simplify the equation:
3x - 3 < 3x - 3
The resulting equation is 3x - 3 < 3x - 3.
Step 3: Subtract 3x from both sides of the inequality to isolate the variable:
3x - 3 - 3x < 3x - 3 - 3x
Simplifying the equation gives -3 < -3.
Step 4: Simplify further:
-3 < -3
At this point, we have -3 < -3, which is not possible. This means that there is no solution to the inequality.
Now, let's graph the solution on a number line:
We will draw a number line and mark an open circle at the point -3. Since there is no solution, we will not shade any portion of the number line. The final graph will consist of just a single open circle at -3.
To solve the inequality 3x - 3 < 3(x - 1), follow these steps:
Step 1: Distribute the 3 on the right side of the inequality:
3x - 3 < 3x - 3.
Step 2: Collect like terms by subtracting 3x from both sides of the inequality:
3x - 3 - 3x < 3x - 3 - 3x.
Simplifying further, we get:
-3 < -3.
Step 3: Simplify the inequality. Since -3 is always less than itself (or any other number), this inequality is true for all real numbers.
Hence, the solution to the inequality 3x - 3 < 3(x - 1) is x ∈ ℝ (all real numbers), and the solution can be represented graphically on a number line by shading the entire line.
Graphically, it would look like this:
-------------●----------------- (with the line fully shaded)
Note: This solution might seem counterintuitive because both sides of the inequality are equal. However, when we solve it, we find that the inequality holds true for all real numbers.