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Chapter 3: Problem 14

Solve each inequality. $$-t>1$$

Short Answer

Expert verified
t < -1

Step by step solution

01

Isolate the variable

To isolate the variable, divide both sides of the inequality by -1. Remember that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign flips.
02

Divide by -1

Divide both sides of the inequality by -1. Doing this gives: \[-t > 1\] becomes \[t < -1\]
03

Write the final solution

After isolating the variable and reversing the inequality, we get \[t < -1\]. This is the solution to the inequality.

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Key Concepts

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

Inequality Sign Reversal
When solving inequalities, one crucial point to remember is that the direction of the inequality sign reverses under certain conditions. This happens when you multiply or divide both sides of the inequality by a negative number.

For example, in the inequality \-t > 1\, the variable \(-t\) is negative. To make the variable positive, you need to divide both sides by \-1\.

When you do this, the inequality sign reverses: \[ -t > 1 \rightarrow t < -1 \]

This is a fundamental rule in solving inequalities and it's essential to getting the correct solution.
Isolate Variable
Isolating the variable is a key step in solving inequalities. It involves manipulating the inequality to get the variable alone on one side of the inequality sign.

For the inequality \-t > 1\, we start by isolating \t\. Since \t\ is negative, we divide both sides by \-1\. This gets us: \[ -t > 1 \rightarrow t < -1 \]

Now, with the variable isolated, you can clearly see the solution to the inequality: \t < -1\.

Isolating the variable simplifies the inequality and helps to find the solution more straightforwardly.
Negative Coefficients
Dealing with negative coefficients in inequalities requires special attention. A coefficient is a number that is multiplied by a variable.

In the inequality \-t > 1\, the variable \t\ has a negative coefficient of \-1\. Negative coefficients complicate the solving process because of the rule of sign reversal when dividing or multiplying by a negative number.

To handle this, we divided both sides of the inequality by \-1\, reversing the inequality sign: \[ -t > 1 \rightarrow t < -1 \]

Understanding how to work with negative coefficients is crucial for accurate solutions in inequalities.

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Most popular questions from this chapter

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