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Part 1
Solve the following inequality. Graph and check your solution.
8z < -32
To solve the inequality 8z < -32, we need to divide both sides of the inequality by 8. However, there is an important distinction to note when dividing inequalities by a negative number.
Since the coefficient of z is positive (8 > 0), we can divide both sides by 8 without changing the direction of the inequality.
8z < -32
Dividing both sides by 8:
z < -32/8
Simplifying:
z < -4
To graph the solution, we need to draw a number line and mark the point -4 with an open circle. This indicates that -4 is not included in the solution. Then we shade the region to the left of -4, which represents all values of z that are less than -4.
Checking the solution:
Let's test a few values to make sure they satisfy the original inequality.
Let z = -5:
8z = 8(-5) = -40
Since -40 < -32, the inequality is true.
Let z = -2:
8z = 8(-2) = -16
Since -16 < -32, the inequality is true.
Let z = 0:
8z = 8(0) = 0
Since 0 is not less than -32, the inequality is false.
Therefore, the solution to the inequality 8z < -32 is z < -4, or in interval notation, (-∞, -4).
To solve the inequality 8z < -32, we need to isolate the variable "z" on one side of the inequality sign.
Step 1: Divide both sides of the inequality by 8. Remember that when you divide or multiply both sides of an inequality by a negative number, you need to reverse the inequality sign.
8z < -32 (Original inequality)
Dividing both sides by 8:
z < -4
Step 2: Graph the solution on a number line. To graph the solution, we mark a point at -4 on the number line and shade all values to the left of it since we are looking for all values of z that are less than -4.
──┬──┬──┬──┬──┬──┬──┬──┬──┬──┬──
... -8 -7 -6 -5 -4 -3 -2 -1 0 1 ...
Step 3: Check your solution. We can choose any value for z less than -4, substitute it into the original inequality, and see if it holds true. Let's choose -5 as an example.
8z < -32 (Original inequality)
8(-5) < -32
-40 < -32
Since -40 is indeed less than -32, the inequality holds true. Hence, z < -4 is the correct solution to the inequality.
Graphically, the solution is any value of z that is to the left of -4 on the number line.
To solve the inequality 8z < -32, we will follow these steps:
Step 1: Divide both sides of the inequality by 8 to isolate the variable z:
8z/8 < -32/8
This simplifies to:
z < -4
Step 2: Now, let's graph the solution on the number line.
On the number line, draw an open circle at -4. Since the inequality is less than (<), we use an open circle to indicate that -4 is not included in the solution set.
Step 3: Shade the region to represent all values of z less than -4.
The shaded region will be to the left of -4 on the number line.
Step 4: Lastly, we need to check our solution. Choose a value of z that is less than -4, such as z = -5.
Substituting z = -5 into the original inequality:
8z < -32
8(-5) < -32
-40 < -32
The solution holds true since -40 is indeed less than -32.
Thus, the solution to the inequality 8z < -32 is z < -4.
Graphically, it can be represented as:
-6 -5 -4 -3 -2 -1 0 1 2 3
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< -- Solution: z < -4