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Chapter 2: Problem 70

Solve each inequality. Graph the solution set, and write it using interval notation. \(2(3 z-5)+4(z+6) \geq 2(3 z+2)+3 z-15\)

Short Answer

Expert verified
The solution is \(z \geq -25\) and the interval notation is \[[-25, \infty)\].

Step by step solution

01

- Distribute

Apply the distributive property to eliminate the parentheses on both sides of the inequality. Left side: \[2(3z - 5) + 4(z + 6) = 6z - 10 + 4z + 24\] Right side: \[2(3z + 2) + 3z - 15 = 6z + 4 + 3z - 15\]
02

- Combine like terms

Combine the like terms on both sides of the inequality. Left side: \[6z - 10 + 4z + 24 = 10z + 14\] Right side: \[6z + 4 + 3z - 15 = 9z - 11\]
03

- Isolate the variable

Isolate the variable z by first subtracting 9z from both sides of the inequality: \[10z + 14 - 9z \geq 9z - 11 - 9z\] This simplifies the inequality to: \[z + 14 \geq -11\]Next, subtract 14 from both sides: \[z + 14 - 14 \geq -11 - 14\] Simplifying this will give: \[z \geq -25\]
04

- Write the solution in interval notation

The solution set in interval notation is: \[[-25, \infty)\]
05

- Graph the solution set

Graph the solution set on a number line. Shade the region to the right of -25 and include a closed circle at -25 to indicate that -25 is part of the solution.```plaintext<----|====================================> -25 鈭� ```

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive property
The distributive property is a fundamental concept in algebra that allows us to eliminate parentheses by distributing a multiplying factor across the terms inside. In our example, we begin by applying the distributive property to the expression:
  • Left side:
    \[2(3z - 5) + 4(z + 6) = 6z - 10 + 4z + 24\]

  • Right side:
    \[2(3z + 2) + 3z - 15 = 6z + 4 + 3z - 15\]
By distributing 2 to both (3z - 5) and 4 to (z + 6) on the left, we simplify step by step. Likewise, distribute 2 to (3z + 2) and then add 3z - 15 on the right. This process helps to eliminate parentheses, preparing us for the next step.
Combine like terms
Combining like terms means to add or subtract terms that have the same variable parts. For example, on both sides of our inequality:
  • Left side:
    \[6z - 10 + 4z + 24 = 10z + 14\]

  • Right side:
    \[6z + 4 + 3z - 15 = 9z - 11\]
We add the coefficients of \(z\) (6z and 4z, and 6z and 3z) together and combine the constant terms (-10 and 24, and 4 and -15). Combining these results leads to simpler expressions on each side.
Isolate the variable
To find the solution for the variable, we need to isolate it on one side of the inequality. Start by subtracting 9z from both sides:
  • \[10z + 14 - 9z \geq 9z - 11 - 9z\]

This simplifies to:
  • \[z + 14 \geq -11\]
Next, subtract 14 from both sides:
  • \[z + 14 - 14 \geq -11 - 14\]
This simplifies to:
  • \[z \geq -25\].
Now our variable, \(z\), stands alone, and we have successfully isolated it.
Interval notation
Interval notation is a way of writing subsets of the real number line. It uses parentheses and brackets to describe sets of numbers. In our case, the solution to the inequality is:\[\z \geq -25\].


  • This is written in interval notation as:
    \[[-25, \infty)\].

  • The bracket [ indicates that -25 is included in the set.

  • The parenthesis ) indicates that infinity is not a specific number but rather, it extends indefinitely.
Interval notation provides a compact, clear way to describe ranges of solutions.
Graphing inequalities
Graphing inequalities helps visualize the range of solutions. On a number line, use a closed circle to indicate a number is included in the solution. Shade the part of the number line that represents all potential values of the variable. For our solution \(z \geq -25\):
  • Place a closed circle at -25.
  • Shade the number line to the right of -25 to show all numbers greater than or equal to -25.

The graph shows all values of z that satisfy the inequality, providing a visual way to understand the solution:

``` plaintext<----|====================================>
-25
鈭� ``` This representation confirms that any number larger than or equal to -25 satisfies the inequality.

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Most popular questions from this chapter

New York and Ohio are among the states with the most remaining drive-in movie screens. New York has 4 more screens than Ohio, and there are 52 screens total in the two states. How many drive-in movie screens remain in each state? (Data from www. drive-ins.com)

The world's largest taco contained approximately \(1 \mathrm{~kg}\) of onion for every \(6.6 \mathrm{~kg}\) of grilled steak. The total weight of these two ingredients was \(617.6 \mathrm{~kg}\). To the nearest tenth of a kilogram, how many kilograms of each ingredient were used to make the taco? (Data from Guinness World Records.

The product of \(5,\) and 3 more than twice a number, is \(85 .\) What is the number?

The difference of a number and 11 is \(-31 .\) What is the number?

If 4 is added to twice a number and this sum is multiplied by \(2,\) the result is the same as if the number is multiplied by 3 and 4 is added to the product. What is the number?
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