Solve the compound inequality.
4≤y+2≤-3(y-2)+24
I don't see the inequality sign.
2<y<7
To solve the compound inequality, let's break it down into two separate inequalities and solve each one step by step.
The given compound inequality is:
4 ≤ y + 2 ≤ -3(y - 2) + 24
Step 1: Solve the first inequality: 4 ≤ y + 2
To isolate the y variable, we need to get rid of the constant term (2) on the right side. We can do this by subtracting 2 from all parts of the inequality:
4 - 2 ≤ y + 2 - 2
2 ≤ y
So, the first inequality is 2 ≤ y.
Step 2: Solve the second inequality: y + 2 ≤ -3(y - 2) + 24
First, simplify the right side of the inequality by applying the distributive property:
y + 2 ≤ -3y + 6 + 24
Combine like terms on the right side:
y + 2 ≤ -3y + 30
Next, we need to isolate y by moving the variable terms to one side and the constant terms to the other side.
Adding 3y to both sides:
y + 3y + 2 ≤ -3y + 3y + 30
4y + 2 ≤ 30
Subtracting 2 from both sides:
4y + 2 - 2 ≤ 30 - 2
4y ≤ 28
Finally, divide both sides by 4 to solve for y:
(4y)/4 ≤ 28/4
y ≤ 7
So, the second inequality is y ≤ 7.
Combining the results of the two inequalities, we have:
2 ≤ y AND y ≤ 7
In interval notation, the solution to the compound inequality is [2, 7].