For the problem |t-4|>t-4 how would you graph it?
To graph the inequality |t-4| > t-4, we can follow these steps:
Step 1: Rewrite the inequality without absolute value notation
Since |t-4| represents the distance between t and 4, we can rewrite the inequality as two separate inequalities, considering the positive and negative distances:
t-4 > t-4 (when t-4 is positive) -> Inequality 1
-(t-4) > t-4 (when t-4 is negative) -> Inequality 2
Step 2: Solve Inequality 1
For Inequality 1, we can simplify it as follows:
t - 4 > t - 4
We can notice that the t terms cancel out, leaving us with:
-4 > -4
Step 3: Analyze Inequality 1
In Inequality 1, -4 > -4 is not a true statement. This means that no matter what value of t we plug in, this inequality is never satisfied. Therefore, there is no solution for the positive distances (t-4 > 0).
Step 4: Solve Inequality 2
For Inequality 2, we distribute the negative sign:
-t + 4 > t - 4
Combine like terms by bringing the t terms to one side:
-2t > -8
Divide by -2, remembering to reverse the inequality sign:
t < 4
Step 5: Analyze Inequality 2
In Inequality 2, t < 4 means that any value of t less than 4 will satisfy the inequality.
Step 6: Plot the graph
On a number line, mark a vertical line at 4 to represent the cut-off point. Shade the region to the left of 4, representing the values of t that satisfy the inequality t < 4.
The graph will have an open circle at 4, indicating that it is not included in the solution, and a shaded region to the left, representing values less than 4.