|y+4|< 1 (1 point)
A: -5<y< -3
B: -3<y<5
C: -4<y<1
D: 1<y<4
To solve the inequality, we can consider two cases: when y+4 is positive and when y+4 is negative.
Case 1: y+4 is positive
In this case, the inequality becomes y+4 < 1. Subtracting 4 from both sides gives y < -3.
Case 2: y+4 is negative
In this case, the inequality becomes -(y+4) < 1. Multiplying both sides by -1 flips the inequality direction, giving y+4 > -1. Subtracting 4 from both sides gives y > -5.
Combining the two cases, we have -5 < y < -3.
Therefore, the correct answer is A: -5 < y < -3
To solve the inequality |y+4| < 1, we can split it into two cases:
Case 1: y+4 < 1
Solving this case, we subtract 4 from both sides:
y < 1 - 4
y < -3
Case 2: -(y+4) < 1
Multiplying both sides by -1 (which changes the direction of the inequality):
y + 4 > -1
Subtracting 4 from both sides:
y > -1 - 4
y > -5
Combining the results from both cases, we get:
-5 < y < -3
Therefore, the answer is option A: -5 < y < -3.
To solve the inequality |y+4| < 1, you need to consider two cases:
Case 1: y+4 is positive (y+4 > 0)
In this case, the absolute value of y+4 is simply y+4. So the inequality becomes y+4 < 1.
To solve this, we subtract 4 from both sides:
y + 4 - 4 < 1 - 4. This simplifies to y < -3.
However, we need to remember that this case is only valid if y+4 > 0. Therefore, we add the condition y+4 > 0 to this solution.
So the solution for this case is y < -3, and y+4 > 0.
Case 2: y+4 is negative (y+4 < 0)
In this case, the absolute value of y+4 is -(y+4), which is equal to -y-4. So the inequality becomes -y-4 < 1.
To solve this, we add 4 to both sides and multiply by -1 to reverse the inequality:
-y - 4 + 4 > 1 + 4. This simplifies to -y > 5.
To solve further, we multiply both sides by -1, which reverses the inequality again:
y < -5.
However, we need to remember that this case is only valid if y+4 < 0. Therefore, we add the condition y+4 < 0 to this solution.
So the solution for this case is y < -5, and y+4 < 0.
To find the combined solution, we need to consider the conditions from both cases:
-3 < y < -5 satisfies both conditions (y+4 > 0 and y+4 < 0), but this is not possible since y cannot be both greater than 0 and less than 0.
Therefore, there is no solution to the given inequality.
In conclusion, none of the options provided (A, B, C, or D) accurately represent the solution to the inequality |y+4| < 1.