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Chapter 1: Problem 44

Fill in the blank to form a correct inequality statement. If \(x>6\) then \(x-6______0\)

Short Answer

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Step by step solution

01

Understand the Given Condition

The problem states that if \(x > 6\), then an inequality involving \(x - 6\) and 0 must hold true.
02

Subtract 6 from Both Sides

Start with the inequality \(x > 6\). Now, subtract 6 from both sides to isolate the expression involving \(x - 6\): \[ x - 6 > 0 \]
03

Interpret the Result

After simplifying, we have the inequality \(x - 6 > 0\). This states that when \(x > 6\), the expression \(x - 6\) will always be greater than 0.

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Key Concepts

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

inequality statement
Inequality statements are used to show that one value is larger or smaller than another value. In our problem, the original statement given is that if \(x > 6\), we need to form a correct inequality involving \(x - 6\) and 0. The purpose of this is to express how much larger \(x\) is than 6. Inequality statements are essential in understanding how values relate to each other, and are commonly used in algebra and numerous applied fields like physics and economics.
subtracting inequalities
One crucial approach to solve inequalities is subtracting the same number from both sides. This method helps maintain the balance of the inequality.
In our exercise, we start with the inequality \(x > 6\). By subtracting 6 from both sides, we can simplify this inequality:
  • Starting inequality: \(x > 6\)
  • Subtract 6 from both sides: \(x - 6 > 0\)

It's important to note that the direction of the inequality sign does not change when you subtract a number. Subtracting inequalities is just like subtracting equations, but we must keep an eye on the inequality sign to ensure it remains correct.
isolating variables
The process of isolating variables is a fundamental skill in solving inequalities and equations. Isolating a variable means you get the variable by itself on one side of the inequality or equation.
In our example, we began with \(x > 6\). By subtracting 6 from both sides, we successfully isolated the variable expression \(x - 6\). Now it is clear that \(x > 6\) directly translates to \(x - 6 > 0\). This isolated variable expression shows us the relationship between \(x\) and a fixed value after subtracting the same number on both sides.
simplifying inequalities
Simplifying inequalities involves reducing them to their most basic form to better understand the relationship between variables and numbers.
In our current example, we started with a more complex form, \(x > 6\).
After subtracting 6 from both sides, we simplified it to: \(x - 6 > 0\). This process made it clear that any value of \(x\) greater than 6 will result in a positive outcome when subtracted by 6. This simplified form, \(x - 6 > 0\), is easier to interpret and use in further calculations or problem-solving steps.

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Most popular questions from this chapter

The inequality \(x^{2}+1<-5\) has no real solution. Explain why.

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