-7+z ¡Ü 3z+7 and 2(z-3)<-4z+2
To solve the inequalities -7+z ≤ 3z+7 and 2(z-3) < -4z+2, we need to isolate the variable z.
Let's begin with the first inequality: -7+z ≤ 3z+7
To isolate z, we can start by subtracting z from both sides of the inequality:
-7+z - z ≤ 3z+7 - z
This simplifies to:
-7 ≤ 2z+7
Next, we can subtract 7 from both sides:
-7 - 7 ≤ 2z+7 - 7
This simplifies to:
-14 ≤ 2z
Now, we divide both sides by 2 to solve for z:
-14/2 ≤ 2z/2
This simplifies to:
-7 ≤ z
Therefore, the solution to the first inequality is z ≥ -7.
Moving on to the second inequality: 2(z-3) < -4z+2
Start by simplifying both sides of the inequality:
2z - 6 < -4z + 2
Next, let's gather the z terms on one side of the inequality and the constant terms on the other side:
2z + 4z < 2 + 6
This simplifies to:
6z < 8
Now, divide both sides by 6 to solve for z:
(6z)/6 < 8/6
This simplifies to:
z < 4/3 or z < 1.33
Therefore, the solution to the second inequality is z < 1.33.
In summary, the solutions to the given inequalities are:
-7 ≤ z and z < 1.33