Solve the inequality 8z+3-2z 51 – Embark on an algebraic adventure as we delve into the intriguing world of inequalities. Today, our spotlight shines upon the enigmatic equation 8z + 3 – 2z ≤ 51. Join us as we unravel the mysteries of simplifying and solving this inequality, unlocking its secrets step by step.

Prepare to witness the power of algebra as we manipulate terms, isolate variables, and uncover the hidden solutions. Along the way, we’ll explore the fascinating concept of graphing solutions, allowing us to visualize the inequality’s boundaries and gain a deeper understanding of its implications.

Simplify the Inequality

To simplify the inequality, we combine like terms on both sides of the inequality sign. Like terms are terms that have the same variable and exponent.

Let’s consider an example to illustrate the simplification process. Suppose we have the inequality 8z + 3 – 2z ≤ 51.

Combining Like Terms

• Combine the like terms 8z and -2z to get 6z.
• Simplify the left-hand side of the inequality: 8z + 3 – 2z = 6z + 3.
• The simplified inequality becomes 6z + 3 ≤ 51.

Isolate the Variable

To isolate the variable, we need to move all the terms containing the variable to one side of the inequality and the constants to the other side.

Step-by-Step Guide

1. Subtract 3 from both sides of the inequality:“`
   

   z+3-2z-3 51-3

   “`

2. Simplify:“`
   

   z 48

   “`

3. Divide both sides of the inequality by 6:“`(6z)/6 48/6“`
4. Simplify:“`z = 8“`

Solve for the Variable: Solve The Inequality 8z+3-2z 51

Now that we have simplified the inequality, we can solve for the variable zto find the solution to the inequality.

To solve for z, we need to isolate it on one side of the inequality sign.

Isolating z

• Subtract 3 from both sides of the inequality:

8 z+ 3 – 3 – 2 z51 – 3

6 z48

• Divide both sides of the inequality by 6:

(6 z)/6 48/6

z8

Graph the Solution

To graph the solution to the inequality 8z + 3 – 2z ≤ 51, we will first solve the inequality to find the range of values for z that satisfy the inequality. Once we have the solution, we will plot these values on a number line and shade the appropriate region to represent the solution set.

Marking the Solution

To mark the solution on the number line, we will use a closed circle at the point where the inequality becomes an equality. In this case, the equality is 8z + 3 – 2z = 51, which simplifies to 6z = 48. Solving for z, we get z = 8. So, we will mark a closed circle at z = 8 on the number line.

Shading the Appropriate Region

Since the inequality is ≤, the solution set includes all values of z that are less than or equal to 8. Therefore, we will shade the region to the left of the closed circle, including the circle itself. This shaded region represents the solution set to the inequality 8z + 3 – 2z ≤ 51.

Example Problem

Let’s solve another inequality problem similar to the previous one:

Solve the inequality: 5x – 2< 18

Simplifying the Inequality, Solve the inequality 8z+3-2z 51

First, let’s simplify the inequality by adding 2 to both sides:

5x

2 + 2< 18 + 2

This gives us:

5x< 20

Isolating the Variable

Next, we want to isolate the variable x on one side of the inequality. To do this, we divide both sides by 5:

(5x) / 5< 20 / 5

This gives us:

x< 4

Solving for the Variable

Finally, we have isolated the variable and solved for it. The solution to the inequality is:

x< 4

Graphing the Solution

To graph the solution, we plot a number line and mark the point x = 4. We then shade the region to the left of this point, since the solution is x< 4.

Clarifying Questions

How do I simplify the inequality 8z + 3- 2z ≤ 51?

Combine like terms to obtain 6z + 3 ≤ 51.

How do I isolate the variable z?

Subtract 3 from both sides of the inequality: 6z ≤ 48.

Divide both sides by 6: z ≤ 8.