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Chapter 1: Problem 6

\(x-3>2\) corresponds to: (a) \(x>3\) (b) \(x>5\) (c) \(x>1\) (d) \(x<3\) (e) \(x>0\).

Short Answer

Expert verified
(b) \(x > 5\).

Step by step solution

01

Understanding the Problem

We need to determine which inequality corresponds to the given inequality: \(x - 3 > 2\).
02

Isolate the Variable

To isolate the variable \(x\), we'll add 3 to both sides of the inequality: \(x - 3 + 3 > 2 + 3\).
03

Simplify the Inequality

Simplifying both sides results in: \(x > 5\).
04

Match with the Given Options

Compare the simplified inequality \(x > 5\) with the provided options. The correct option is (b) \(x > 5\).

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Key Concepts

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

solving inequalities
When solving inequalities, it's important to follow similar steps as you would for solving equations. However, you need to pay extra attention to the inequality signs. For the given problem, the inequality is initially presented as:

\(x - 3 > 2\)

The goal is to determine which inequality matches this original statement once it is simplified.

Follow each step methodically, making sure to perform the same operation on both sides to maintain the balance.
isolation of variables
Isolation of the variable is a crucial step in simplifying inequalities. Here, the aim is to have the variable on one side of the inequality to determine the range of its possible values.

Take the inequality \(x - 3 > 2\) as an example. The first step to isolate the variable \(x\) involves removing the constant term from the left side.

Add 3 to both sides:

\(x - 3 + 3 > 2 + 3\)

This simplifies to \(x > 5\) effectively isolating \(x\) on one side.
simplifying inequalities
Simplifying inequalities can often make them easier to understand and solve. After isolating the variable in the inequality \(x - 3 > 2\), the next step is to simplify the expression.

Here, adding 3 simplifies the inequality to:

\(x > 5\)

This means that \(x\) must be greater than 5. Simplifying makes it clearer and easier to see the relationship between \(x\) and the numbers it is compared to.
comparison of inequalities
Once simplified, the final step is to compare the simplified inequality with the given options. In this exercise, the correct simplified inequality \(x > 5\) matches option (b).

Comparing inequalities helps ensure that the solution accurately represents the original problem statement. This process involves:
  • Ensuring both sides have been simplified properly
  • Confirming the direction of the inequality sign
  • Matching it with provided options
By properly comparing inequalities, you validate your final answer and ensure it is correctly derived.

So, in this case, option (b) \(x > 5\) is indeed the correct match.

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Most popular questions from this chapter

Given that the roots of the equation \(a x^{2}+b x+c=0 \quad\) are \(\beta\) and \(n \beta\), show that \((n+1)^{2} a c=n b^{2}\) (U of \(\mathrm{L}\) )

If the roots of \(x^{2}+p x+q=0\) are \(\alpha\) and \(\beta\), where \(\alpha\) and \(\beta\) are nonzero, form the equation whose roots are \(\frac{2}{\alpha}, \frac{2}{\beta}\). \((\mathrm{U}\) of \(\mathrm{L}) \mathrm{p}\)

If \(x^{2}+4 x+p \equiv(x+q)^{2}+1\), the values of \(p\) and \(q\) are: (a) \(p=5, q=2\) (b) \(p=1, q=2\) (c) \(p=2, q=5\) (d) \(p=-1, q=5\) (e) \(p=0, q=-1\).

Express \(\frac{9 x}{(2 x+1)^{2}(1-x)}\) as a sum of partial fractions with constant numerators. \((\mathrm{U}\) of \(\mathrm{L}) \mathrm{p}\)

7) \(f(x) \equiv x^{2}+\frac{1}{x}+1\) corresponds to: (a) \(f(1)=1\) (b) \(f(-1)=3\) (c) \(f(0)=1\) (d) \(f(1)=3\) (e) \(f(-1)=-1\).
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