Solve the inequality graph solution 8| x+3/4|<2answers are ,x< -1or x>-1/2, -5/32<x<5/32, -1<x<-1/2, x<-5/32or x>5/32 thank you
8| x+3/4|< 2
| x+3/4| < 1/4
x+3/4 < 1/4 AND -x-3/4 < 1/4
x < -1/2 AND -x < 1
x < -1/2 AND x > -1
which matches:
-1 < x < -1/2
To solve the inequality 8 |x + 3/4| < 2, we will break it down into two separate cases:
Case 1: x + 3/4 ≥ 0
In this case, the absolute value of x + 3/4 will remain unchanged. So the inequality becomes:
8(x + 3/4) < 2
Simplifying this inequality, we get:
8x + 6 < 2
Then, subtracting 6 from both sides, we get:
8x < -4
Dividing both sides by 8, we find:
x < -1/2
Therefore, for the case x + 3/4 ≥ 0, the solution is:
x < -1/2
Case 2: x + 3/4 < 0
In this case, the absolute value of x + 3/4 will be multiplied by -1, resulting in:
8(-x - 3/4) < 2
Simplifying this inequality, we get:
-8x - 6 < 2
Adding 6 to both sides, we find:
-8x < 8
Dividing both sides by -8 (and flipping the inequality sign), we have:
x > -1
Therefore, for the case x + 3/4 < 0, the solution is:
x > -1
Combining the solutions from both cases, we have:
x < -1/2 or x > -1
So, the solution to the given inequality, 8 |x + 3/4| < 2, is x < -1/2 or x > -1.