Question
Solve the compound inequality. \[ 4 y+4 \geq 0 \text { or } 2 y-4>2 \] Write the solution in interval notation: If there is no solution, enter \( \varnothing \). (ם, \( \square) \)
Ask by Mcfarlane Lyons. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is \( y \geq -1 \), which in interval notation is \([-1, \infty)\).
Solution
To solve the compound inequality
4y + 4 ≥ 0 or 2y - 4 > 2
we solve each inequality separately:
1. For 4y + 4 ≥ 0:
Subtract 4 from both sides:
4y ≥ -4
Divide both sides by 4:
y ≥ -1
2. For 2y - 4 > 2:
Add 4 to both sides:
2y > 6
Divide both sides by 2:
y > 3
Since the compound inequality uses "or," we take the union of the two solution sets. The solution y ≥ -1 already includes all numbers greater than 3, so the union is simply:
y ≥ -1
In interval notation, the solution is:
[-1, ∞)
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Simplify this solution Beyond the Answer
To solve the inequalities, we'll tackle them one at a time. 1. For the first inequality, \( 4y + 4 \geq 0 \): \[ 4y \geq -4 \] Dividing both sides by 4: \[ y \geq -1 \] 2. For the second inequality, \( 2y - 4 > 2 \): \[ 2y > 6 \] Dividing both sides by 2: \[ y > 3 \] Now, we combine these results since this is an "or" statement. The solution to the compound inequality is: - \( y \geq -1 \): This means any number greater than or equal to -1 is part of the solution. - \( y > 3 \): This means any number greater than 3 is part of the solution as well. In interval notation, the solution is: \[ [-1, \infty) \]
