Which Are Correct Representations Of The Inequality 3(2x 5) < 5(2 X)? Select Two Options.

Which Are Correct Representations Of The Inequality 3(2x 5) < 5(2 X)? Select Two Options.

Understanding inequalities is fundamental in algebra, as they allow us to compare quantities and determine ranges of solutions. When given an inequality such as 3(2x - 5) < 5(2x), it is crucial to interpret and manipulate the expression correctly to find its valid representations. Selecting the correct forms of this inequality requires careful algebraic processing, attention to sign changes, and awareness of how inequalities behave under various operations. In this article, we explore the step-by-step process of transforming the given inequality, analyze multiple potential representations, and identify which options accurately reflect the original inequality.

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Understanding the Original Inequality

Breaking Down the Expression

The initial inequality is:

```plaintext
3(2x - 5) < 5(2x)
```

This expression involves algebraic terms with coefficients and variables, and the goal is to simplify and interpret the inequality correctly.

Expanding the Terms

Applying the distributive property to both sides:

• Left side: 3(2x - 5) = 3 2x - 3 5 = 6x - 15
• Right side: 5(2x) = 10x

Thus, the inequality simplifies to:

```plaintext
6x - 15 < 10x
```

Isolating the Variable

To better understand the solution set, we need to isolate x:

Subtract 6x from both sides:

```plaintext
6x - 15 - 6x < 10x - 6x
```

which simplifies to:

```plaintext
-15 < 4x
```

Then, divide both sides by 4:

```plaintext
\frac{-15}{4} < x
```

Alternatively, written as:

```plaintext
x > -\frac{15}{4}
```

This is the solution in its simplest form: x is greater than -3.75.

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Representations of the Inequality

The simplified form, x > -15/4, is a fundamental representation. However, multiple algebraic expressions can represent the same inequality, especially when considering equivalent transformations. The question asks for two correct representations of the original inequality, so we need to analyze various forms and determine which are valid.

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Option 1: The Simplified Form x > -\frac{15}{4}

This straightforward representation is derived directly from the algebraic steps:

• Starting from the expanded inequality: 6x - 15 < 10x


• Subtracting 6x: -15 < 4x


• Dividing both sides by 4: x > -15/4

Why is this correct?

• Each step is valid because we performed the same operation on both sides.


• Dividing by a positive number (4) maintains the inequality sign.


• The final form clearly states the set of x-values satisfying the original inequality.

Conclusion: This is a correct and direct representation of the original inequality.

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Option 2: Rewriting as 3(2x - 5) < 5(2x)

This form is the original inequality itself.

Why is this correct?

• It accurately reflects the problem statement before any algebraic manipulation.


• It maintains all components of the inequality.

Conclusion: The original inequality is always a valid representation of itself; thus, it is correct.

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Other Potential Forms and Their Validity

While the above two are straightforward, other forms might be proposed, and their correctness depends on whether they are algebraically equivalent to the original inequality.

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Analyzing Other Possible Representations

Option 3: The inequality written as 6x - 15 < 10x

Derivation:

• From the original expression, we expanded to get 6x - 15 < 10x.


• This form is a direct algebraic form before isolating x.

Validity:

• It is equivalent to the original inequality since it results from valid expansion.


• Therefore, this is also an acceptable representation.

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Option 4: The inequality written as 4x > -15

Derivation:

• From previous steps: -15 < 4x


• Reversing the inequality: 4x > -15

Validity:

• Since the inequality sign was reversed when dividing both sides by a positive number (4), this is valid.


• It is equivalent to the previous form x > -15/4.

Conclusion: This is a correct representation.

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Option 5: The inequality expressed as x < -\frac{15}{4}

Analysis:

• This is the inverse of the correct solution x > -15/4.


• It would only be correct if the inequality sign were reversed, which it isn't.

Validity:

• It is incorrect because the original inequality indicates x is greater than -15/4, not less.

---

Summary of Correct Representations

Based on the algebraic steps and transformations, the two correct representations are:

1. x > -15/4


2. 6x - 15 < 10x

These two forms are directly derived or equivalent to the original inequality and preserve the inequality's meaning.

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Conclusion: Selecting the Two Correct Options

In multiple-choice settings, options often include various algebraic forms. The two correct representations of the inequality 3(2x - 5) < 5(2x), based on the analysis above, are:

• The simplified form: x > -15/4


• The expanded form before isolating x: 6x - 15 < 10x

Both are algebraically equivalent and valid, confirming their correctness.

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Final Remarks

Understanding how to manipulate inequalities correctly is essential for solving and representing them accurately. Always verify each step, especially when multiplying or dividing by negative numbers (which would reverse the inequality sign). Recognizing equivalent forms helps in simplifying and solving inequalities efficiently. In this case, the key was expanding, simplifying, and understanding the relationship between the original and derived inequalities.

Mastering these transformations ensures clarity in algebraic reasoning and prepares you for more complex inequality problems in mathematics.