Solve the inequality. If there is no solution, write no solution.
|2x-7|>1
show your work
The inequality is |2x-7| > 1.
To solve this inequality, we need to consider two cases:
1) 2x-7 > 1
2) 2x-7 < -1
Let's solve the first case: 2x - 7 > 1.
Adding 7 to both sides:
2x > 8
Dividing both sides by 2 (remember to flip the inequality symbol when dividing by a negative number):
x > 4
Now let's solve the second case: 2x - 7 < -1.
Adding 7 to both sides:
2x < 6
Dividing both sides by 2:
x < 3
Therefore, the solution to the inequality |2x-7| > 1 is x < 3 or x > 4.
To solve the inequality |2x-7| > 1, we will solve it in two parts: when the expression inside the absolute value is positive, and when it is negative.
Part 1: When 2x-7 is positive:
When the expression inside the absolute value is positive, the absolute value signs can be removed, so we have:
2x - 7 > 1
Adding 7 to both sides of the inequality, we get:
2x > 8
Finally, dividing both sides of the inequality by 2, we have:
x > 4
Part 2: When 2x-7 is negative:
When the expression inside the absolute value is negative, the absolute value signs can also be removed, but the inequality sign must be reversed. So we have:
-(2x - 7) > 1
Expanding the negation, we get:
-2x + 7 > 1
Subtracting 7 from both sides of the inequality, we have:
-2x > -6
Dividing both sides of the inequality by -2, we get:
x < 3
Combining the solutions from both parts, we have:
x > 4 OR x < 3
Therefore, the solution to the inequality |2x-7| > 1 is x > 4 OR x < 3.
To solve the inequality |2x-7|>1, we need to consider two cases:
Case 1: (2x-7) > 1
In this case, we don't need to take the absolute value because (2x-7) is already positive. Solving for x:
2x - 7 > 1
2x > 1 + 7
2x > 8
x > 4
Case 2: -(2x-7) > 1
In this case, we need to negate (2x-7) because it is negative. Solving for x:
-(2x - 7) > 1
-2x + 7 > 1
-2x > 1 - 7
-2x > -6
x < 3
Combining the solutions from both cases, we find that x is greater than 4 or less than 3. Therefore, the solution to the inequality |2x-7|>1 is: x < 3 or x > 4.