Two-Step InequalitiesIf $x \ \textgreater \ 3$, Which Of The Following Values Can Be Used To Verify The Solution $7 \cdot 5 - 3 \ \textgreater \ 18$?Given: $35 - 3$32 \ \textgreater \ 18$Possible Values For

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Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more values. A two-step inequality is a type of inequality that involves two operations, such as addition and multiplication, or subtraction and division. In this article, we will explore the concept of two-step inequalities and provide a step-by-step guide on how to solve them.

What are Two-Step Inequalities?

A two-step inequality is an inequality that involves two operations, such as addition and multiplication, or subtraction and division. For example, consider the inequality 7⋅5−3 \textgreater 187 \cdot 5 - 3 \ \textgreater \ 18. This inequality involves two operations: multiplication and subtraction.

Types of Two-Step Inequalities

There are two types of two-step inequalities:

  • Addition and Multiplication: This type of inequality involves addition and multiplication operations. For example, 2x+5 \textgreater 102x + 5 \ \textgreater \ 10.
  • Subtraction and Division: This type of inequality involves subtraction and division operations. For example, x−3 \textless 5x - 3 \ \textless \ 5.

How to Solve Two-Step Inequalities

To solve a two-step inequality, follow these steps:

  1. Simplify the inequality: Simplify the inequality by combining like terms.
  2. Isolate the variable: Isolate the variable by performing the inverse operation of the operation that is being performed on the variable.
  3. Check the solution: Check the solution by plugging it back into the original inequality.

Example 1: Solving an Addition and Multiplication Inequality

Consider the inequality 2x+5 \textgreater 102x + 5 \ \textgreater \ 10. To solve this inequality, follow these steps:

  1. Simplify the inequality: Simplify the inequality by combining like terms. 2x+5 \textgreater 102x + 5 \ \textgreater \ 10 becomes 2x \textgreater 52x \ \textgreater \ 5.
  2. Isolate the variable: Isolate the variable by performing the inverse operation of the operation that is being performed on the variable. Divide both sides of the inequality by 2. 2x \textgreater 52x \ \textgreater \ 5 becomes x \textgreater 2.5x \ \textgreater \ 2.5.
  3. Check the solution: Check the solution by plugging it back into the original inequality. Plug in x=3x = 3 into the original inequality. 2(3)+5 \textgreater 102(3) + 5 \ \textgreater \ 10 becomes 6+5 \textgreater 106 + 5 \ \textgreater \ 10, which is true.

Example 2: Solving a Subtraction and Division Inequality

Consider the inequality x−3 \textless 5x - 3 \ \textless \ 5. To solve this inequality, follow these steps:

  1. Simplify the inequality: Simplify the inequality by combining like terms. x−3 \textless 5x - 3 \ \textless \ 5 becomes x \textless 8x \ \textless \ 8.
  2. Isolate the variable: Isolate the variable by performing the inverse operation of the operation that is being performed on the variable. Add 3 to both sides of the inequality. x \textless 8x \ \textless \ 8 becomes x+3 \textless 11x + 3 \ \textless \ 11.
  3. Check the solution: Check the solution by plugging it back into the original inequality. Plug in x=4x = 4 into the original inequality. 4−3 \textless 54 - 3 \ \textless \ 5 becomes 1 \textless 51 \ \textless \ 5, which is true.

Conclusion

In conclusion, two-step inequalities are a fundamental concept in mathematics that deals with the comparison of two or more values. By following the steps outlined in this article, you can solve two-step inequalities and verify the solution.

Possible Values for Verification

Given the inequality 7⋅5−3 \textgreater 187 \cdot 5 - 3 \ \textgreater \ 18, which of the following values can be used to verify the solution?

  • 35 - 3: This value can be used to verify the solution because it is equal to 7â‹…5−37 \cdot 5 - 3.
  • 32 \ \textgreater \ 18: This value is not equal to 7â‹…5−37 \cdot 5 - 3 and cannot be used to verify the solution.

Discussion

Two-step inequalities are a fundamental concept in mathematics that deals with the comparison of two or more values. By following the steps outlined in this article, you can solve two-step inequalities and verify the solution.

References

Keywords

  • Two-step inequalities
  • Addition and multiplication
  • Subtraction and division
  • Inequalities
  • Mathematics
    Two-Step Inequalities: A Comprehensive Guide =====================================================

Q&A: Two-Step Inequalities

Q: What is a two-step inequality?

A: A two-step inequality is an inequality that involves two operations, such as addition and multiplication, or subtraction and division.

Q: What are the types of two-step inequalities?

A: There are two types of two-step inequalities:

  • Addition and Multiplication: This type of inequality involves addition and multiplication operations. For example, 2x+5 \textgreater 102x + 5 \ \textgreater \ 10.
  • Subtraction and Division: This type of inequality involves subtraction and division operations. For example, x−3 \textless 5x - 3 \ \textless \ 5.

Q: How do I solve a two-step inequality?

A: To solve a two-step inequality, follow these steps:

  1. Simplify the inequality: Simplify the inequality by combining like terms.
  2. Isolate the variable: Isolate the variable by performing the inverse operation of the operation that is being performed on the variable.
  3. Check the solution: Check the solution by plugging it back into the original inequality.

Q: What is the difference between a two-step inequality and a one-step inequality?

A: A one-step inequality involves only one operation, such as addition or subtraction. A two-step inequality involves two operations, such as addition and multiplication, or subtraction and division.

Q: Can I use a calculator to solve a two-step inequality?

A: Yes, you can use a calculator to solve a two-step inequality. However, it is always a good idea to check the solution by plugging it back into the original inequality.

Q: What if I have a two-step inequality with a variable on both sides?

A: If you have a two-step inequality with a variable on both sides, you can isolate the variable by performing the inverse operation of the operation that is being performed on the variable.

Q: Can I use a graphing calculator to solve a two-step inequality?

A: Yes, you can use a graphing calculator to solve a two-step inequality. Graphing calculators can help you visualize the solution to the inequality.

Q: What if I have a two-step inequality with a fraction?

A: If you have a two-step inequality with a fraction, you can simplify the fraction by finding the least common denominator (LCD).

Q: Can I use a computer algebra system (CAS) to solve a two-step inequality?

A: Yes, you can use a CAS to solve a two-step inequality. CAS systems can help you solve inequalities and other mathematical problems.

Q: What if I have a two-step inequality with a negative number?

A: If you have a two-step inequality with a negative number, you can simplify the inequality by multiplying both sides by -1.

Q: Can I use a word problem to solve a two-step inequality?

A: Yes, you can use a word problem to solve a two-step inequality. Word problems can help you apply the concept of two-step inequalities to real-world situations.

Conclusion

In conclusion, two-step inequalities are a fundamental concept in mathematics that deals with the comparison of two or more values. By following the steps outlined in this article, you can solve two-step inequalities and verify the solution.

References

Keywords

  • Two-step inequalities
  • Addition and multiplication
  • Subtraction and division
  • Inequalities
  • Mathematics