Answer:
1. -3 < 2x - 5 < 7
To solve this inequality, we will split it into two separate inequalities:
-3 < 2x - 5 (Inequality 1)
2x - 5 < 7 (Inequality 2)
Now, let's solve each inequality separately:
Inequality 1:
-3 < 2x - 5
We add 5 to both sides of the inequality to isolate the term with 2x:
-3 + 5 < 2x
2 < 2x
Now, divide both sides by 2 to solve for x:
2/2 < x
1 < x
So, the solution to Inequality 1 is x > 1.
Inequality 2:
2x - 5 < 7
We add 5 to both sides of the inequality:
2x - 5 + 5 < 7 + 5
2x < 12
Next, divide both sides by 2:
(2x)/2 < 12/2
x < 6
So, the solution to Inequality 2 is x < 6.
Combining the solutions to both inequalities, we have: 1 < x < 6.
Now, let's move on to the inequality you provided in text form:
2. 1 < 4 - x < 6
Similar to before, let's split it into two separate inequalities:
1 < 4 - x (Inequality 3)
4 - x < 6 (Inequality 4)
Solving each inequality:
Inequality 3:
1 < 4 - x
Subtract 4 from both sides:
1 - 4 < -x
-3 < -x
We multiply both sides by -1 to reverse the inequality:
-1*(-3) > x
3 > x
So, the solution to Inequality 3 is x < 3.
Inequality 4:
4 - x < 6
Subtract 4 from both sides:
4 - 4 - x < 6 - 4
-x < 2
Multiply both sides by -1 to reverse the inequality:
-1 * (-x) > -1 * 2
x > -2
So, the solution to Inequality 4 is x > -2.
Combining the solutions to both inequalities, we have: x > -2 and x < 3.
I hope this helps! Let me know if you have any further questions.