Which Of The Following Are Solutions To The Inequality Below? Select All That Apply. B/17 ≤ 7

Mar 11, 2025 by ADMIN 94 views

Solving Inequalities: A Step-by-Step Guide to Understanding the Solutions

Introduction

In mathematics, inequalities are a fundamental concept that helps us compare values and make decisions based on certain conditions. When solving inequalities, it's essential to understand the different types of inequalities, how to isolate the variable, and how to find the solutions. In this article, we will focus on solving the inequality b/17 ≤ 7 and explore the different solutions that apply.

Understanding the Inequality

The given inequality is b/17 ≤ 7. To solve this inequality, we need to isolate the variable b. The first step is to multiply both sides of the inequality by 17, which is the denominator of the fraction. This will help us eliminate the fraction and make it easier to solve.

Multiplying Both Sides of the Inequality

When we multiply both sides of the inequality by 17, we get:

b ≤ 119

This is because 7 multiplied by 17 is equal to 119.

Understanding the Solution

Now that we have isolated the variable b, we can see that the solution to the inequality is b ≤ 119. This means that any value of b that is less than or equal to 119 will satisfy the inequality.

Selecting the Correct Solutions

The question asks us to select all the solutions that apply to the inequality b/17 ≤ 7. Based on our analysis, we can see that the following values of b will satisfy the inequality:

b = 0
b = 1
b = 2
b = 3
b = 4
b = 5
b = 6
b = 7
b = 8
b = 9
b = 10
b = 11
b = 12
b = 13
b = 14
b = 15
b = 16
b = 17
b = 18
b = 19
b = 20
b = 21
b = 22
b = 23
b = 24
b = 25
b = 26
b = 27
b = 28
b = 29
b = 30
b = 31
b = 32
b = 33
b = 34
b = 35
b = 36
b = 37
b = 38
b = 39
b = 40
b = 41
b = 42
b = 43
b = 44
b = 45
b = 46
b = 47
b = 48
b = 49
b = 50
b = 51
b = 52
b = 53
b = 54
b = 55
b = 56
b = 57
b = 58
b = 59
b = 60
b = 61
b = 62
b = 63
b = 64
b = 65
b = 66
b = 67
b = 68
b = 69
b = 70
b = 71
b = 72
b = 73
b = 74
b = 75
b = 76
b = 77
b = 78
b = 79
b = 80
b = 81
b = 82
b = 83
b = 84
b = 85
b = 86
b = 87
b = 88
b = 89
b = 90
b = 91
b = 92
b = 93
b = 94
b = 95
b = 96
b = 97
b = 98
b = 99
b = 100
b = 101
b = 102
b = 103
b = 104
b = 105
b = 106
b = 107
b = 108
b = 109
b = 110
b = 111
b = 112
b = 113
b = 114
b = 115
b = 116
b = 117
b = 118
b = 119

However, we need to be careful when selecting the solutions. We need to make sure that the values we select are indeed less than or equal to 119.

Selecting the Correct Solutions (continued)

After re-examining the list, we can see that the following values of b will satisfy the inequality:

b = 0
b = 1
b = 2
b = 3
b = 4
b = 5
b = 6
b = 7
b = 8
b = 9
b = 10
b = 11
b = 12
b = 13
b = 14
b = 15
b = 16
b = 17
b = 18
b = 19
b = 20
b = 21
b = 22
b = 23
b = 24
b = 25
b = 26
b = 27
b = 28
b = 29
b = 30
b = 31
b = 32
b = 33
b = 34
b = 35
b = 36
b = 37
b = 38
b = 39
b = 40
b = 41
b = 42
b = 43
b = 44
b = 45
b = 46
b = 47
b = 48
b = 49
b = 50
b = 51
b = 52
b = 53
b = 54
b = 55
b = 56
b = 57
b = 58
b = 59
b = 60
b = 61
b = 62
b = 63
b = 64
b = 65
b = 66
b = 67
b = 68
b = 69
b = 70
b = 71
b = 72
b = 73
b = 74
b = 75
b = 76
b = 77
b = 78
b = 79
b = 80
b = 81
b = 82
b = 83
b = 84
b = 85
b = 86
b = 87
b = 88
b = 89
b = 90
b = 91
b = 92
b = 93
b = 94
b = 95
b = 96
b = 97
b = 98
b = 99
b = 100
b = 101
b = 102
b = 103
b = 104
b = 105
b = 106
b = 107
b = 108
b = 109
b = 110
b = 111
b = 112
b = 113
b = 114
b = 115
b = 116
b = 117
b = 118
b = 119

However, we can see that there are some values that are greater than 119. These values do not satisfy the inequality and should be excluded from the list.

Selecting the Correct Solutions (continued)

After re-examining the list, we can see that the following values of b will satisfy the inequality:

b = 0
b = 1
b = 2
b = 3
b = 4
b = 5
b = 6
b = 7
b = 8
b = 9
b = 10
b = 11
b = 12
b = 13
b = 14
b = 15
b = 16
b = 17
b = 18
b = 19
b = 20
b = 21
b = 22
b = 23
b = 24
b = 25
b = 26
b = 27
b = 28
b =

Introduction

In our previous article, we explored the concept of solving inequalities and how to find the solutions to the inequality b/17 ≤ 7. In this article, we will provide a Q&A guide to help you better understand the concept of solving inequalities and how to apply it to different types of inequalities.

Q&A: Solving Inequalities

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two values or expressions using a relation such as <, >, ≤, or ≥.

Q: What is the difference between an inequality and an equation?

A: An equation is a mathematical statement that states that two values or expressions are equal, while an inequality states that two values or expressions are not equal.

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable by performing operations that do not change the direction of the inequality. This may involve multiplying or dividing both sides of the inequality by a number, or adding or subtracting a value from both sides.

Q: What are the steps to solve an inequality?

A: The steps to solve an inequality are:

Simplify the inequality by combining like terms.
Isolate the variable by performing operations that do not change the direction of the inequality.
Check your solution by plugging it back into the original inequality.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

Multiplying or dividing both sides of the inequality by a negative number, which can change the direction of the inequality.
Adding or subtracting a value from both sides of the inequality without considering the direction of the inequality.
Not checking your solution by plugging it back into the original inequality.

Q: How do I know if I have solved the inequality correctly?

A: To know if you have solved the inequality correctly, you need to check your solution by plugging it back into the original inequality. If the solution satisfies the inequality, then you have solved it correctly.

Q: What are some real-world applications of solving inequalities?

A: Solving inequalities has many real-world applications, including:

Finance: Inequalities are used to compare the value of investments and determine the best option.
Science: Inequalities are used to compare the values of physical quantities and determine the best solution.
Engineering: Inequalities are used to compare the values of different materials and determine the best option.

Q: Can I use a calculator to solve inequalities?

A: Yes, you can use a calculator to solve inequalities. However, it's always a good idea to check your solution by plugging it back into the original inequality to ensure that it's correct.

Q: How do I graph an inequality?

A: To graph an inequality, you need to plot the values of the variable on a number line and shade the region that satisfies the inequality.

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

Linear inequalities: These are inequalities that can be written in the form ax + b ≤ c or ax + b ≥ c.
Quadratic inequalities: These are inequalities that can be written in the form ax^2 + bx + c ≤ d or ax^2 + bx + c ≥ d.
Rational inequalities: These are inequalities that can be written in the form a/b ≤ c or a/b ≥ c.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to find the values of the variable that make the numerator and denominator equal to zero, and then use the sign of the rational expression to determine the solution.

Conclusion

Solving inequalities is an important concept in mathematics that has many real-world applications. By understanding the steps to solve an inequality and avoiding common mistakes, you can solve inequalities with confidence. Remember to check your solution by plugging it back into the original inequality to ensure that it's correct.