i'm doing absolute value in open sentences and then graphing them. how would you solve |t-4| > t-4 ?
it could be a no solution answer??
+(t-4) > t-4
0>0 , no solution
or
-(t-4) > t-4
-t+4 >t-4
-2t > -8
t < 4
check with some numbers
t = 1 , which is less than 4
|1-4| > 1-4
3 > -3 , YUP
t = -50 , which is less than 4
|-50-4| > -50 - 4
54 > -54 , YUP
let t = 6 , which is greater than 4
|6-4| > 6-4
2 > 2 , no, as it should be
looks like t > 4 is my solution
To solve |t-4| > t-4, you can follow these steps:
Step 1: Remove the absolute value brackets by considering two cases:
Case 1: (t-4) is positive or zero
In this case, the absolute value |t-4| is the same as (t-4).
So, the inequality becomes (t-4) > t-4.
Case 2: (t-4) is negative
In this case, the absolute value |t-4| is the negative of (t-4), which is -(t-4) or 4-t.
So, the inequality becomes 4-t > t-4.
Step 2: Solve the inequalities separately for each case:
For Case 1: (t-4) > t-4
Start by simplifying both sides:
t-4 > t-4
This inequality shows that the two sides are equal, which means that the inequality holds true for all real values of t. In other words, it is always true. Thus, all values of t satisfy this inequality.
For Case 2: 4-t > t-4
Start by simplifying both sides:
-1(t-4) > t-4
Note that multiplying both sides by -1 changes the direction of the inequality.
t-4 < 4-t
Now, let's solve this inequality:
Add t to both sides to isolate the t terms on one side:
t + (t-4) < 4
Combine like terms:
2t - 4 < 4
Add 4 to both sides:
2t < 8
Divide both sides by 2 to solve for t:
t < 4
Step 3: Combine the solutions from both cases:
In Case 1, we found that all values of t satisfy the inequality.
In Case 2, we found that t must be less than 4.
Thus, the combined solution is t < 4.