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Chapter 3: Problem 13

Solve each inequality. $$-x<4$$

Short Answer

Expert verified
The solution is \(x > -4\).

Step by step solution

01

Isolate the variable

The inequality given is \(-x < 4\). To isolate \(x\), both sides of the inequality need to be divided by -1. Remember that when dividing or multiplying both sides of an inequality by a negative number, the inequality sign reverses direction.
02

Divide by -1

Divide both sides of the inequality by -1 to isolate \(x\): \[ -x < 4 \implies x > -4 \]

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Key Concepts

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

Isolating the Variable
To solve inequalities like \(-x < 4\), the first step is to isolate the variable. Here, our variable is \(x\). To isolate \(x\), we need to transform the inequality so that \(x\) stands alone on one side. In this case, \(-x < 4\), \(x\) is currently multiplied by -1. Hence, we will divide both sides by -1 in the next step.
By isolating the variable, you make it easier to see what values satisfy the inequality. Make sure to follow the rules carefully when working with inequalities.
Dividing by Negative Numbers
Dividing by negative numbers is trickier in inequalities than with equalities. The key rule is that when you divide or multiply both sides of an inequality by a negative number, you need to reverse the inequality sign.聽
For example, in \(-x < 4\), we divide both sides by -1: \[ \frac{-x}{-1} > \frac{4}{-1} \implies x > -4 \]
Notice how the '<' sign flips to '>' after this operation. This step is crucial because it ensures the inequality maintains its truth when the direction changes.
Reversing Inequality Sign
Reversing the inequality sign is a vital step when dealing with negative numbers in inequalities. It keeps the inequality accurate when you multiply or divide by negative numbers.
In the example \(-x < 4\), once you isolate \(x\) by dividing by -1, you must reverse the inequality sign. This changes < to >, providing the correct solution: \ x > -4 \.
So always remember: divide or multiply by a negative number, flip the inequality sign! It鈥檚 like flipping a seesaw; the balance point changes direction to maintain equilibrium.

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