Problem 102 Solve for x. $$5-2 x \geq 17$$... [FREE SOLUTION]

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Understanding Elementary Algebra with Geometry

Lewis Hirsch, Arthur Goodman

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 7: Problem 102

Solve for x. $$5-2 x \geq 17$$

Short Answer

Expert verified

x 鈮� -6

Step by step solution

Isolate the variable term

To isolate the term involving x, we need to move the constant on the left side to the right side of the inequality. Subtract 5 from both sides:\[ 5 - 2x - 5 \geq 17 - 5 \]This simplifies to:\[ -2x \geq 12 \]

Solve for x

Divide both sides of the inequality by -2. Remember to reverse the inequality sign when dividing both sides by a negative number:\[ \frac{-2x}{-2} \leq \frac{12}{-2} \]This simplifies to:\[ x \leq -6 \]

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Key Concepts

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

Isolation of Variable

When solving inequalities, one of the key steps is isolating the variable. This simply means we want the variable (in this case, x) on one side of the inequality by itself.

For example, in the inequality: \[ 5 - 2x \geq 17 \]

The term with x is \-2x\. To isolate it, we start by moving the constant term (which is 5) from the left side to the right side of the inequality.

We do this by subtracting 5 from both sides:
\[ 5 - 2x - 5 \geq 17 - 5 \]

This simplifies to: \[ -2x \geq 12 \]

Isolating the variable helps us focus on the term containing x, making it easier to solve the inequality.

Inequality Reversal

When working with inequalities, it's crucial to remember a key rule: whenever we multiply or divide both sides of an inequality by a negative number, we must reverse the inequality sign.

Let's consider the previous result: \[ -2x \geq 12 \]

To solve for x, we need to divide both sides by -2. However, dividing by a negative number reverses the inequality:
\[ \frac{-2x}{-2} \leq \frac{12}{-2} \]

Here, the \geq\ sign changes to a \leq\ sign, resulting in: \[ x \leq -6 \]

Failing to reverse the sign can lead to incorrect solutions, so always remember this important step.

Dividing by Negative Numbers

Dividing by negative numbers in inequalities is a special aspect that requires extra attention.

In the final step of our example: \[ -2x \geq 12 \]

We divided both sides by -2: \[ \frac{-2x}{-2} \leq \frac{12}{-2} \]

The division changes the inequality sign from \geq\ to \leq\. This reversal occurs because negative numbers flip the order of the inequality when multiplied or divided.

As a general rule, whenever we multiply or divide both sides of an inequality by a negative number, we must flip the inequality sign.
By keeping this rule in mind, we ensure our solutions are correct and accurately reflect the relationship between the variables.

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91影视

Solve for x. $$5-2 x \geq 17$$

Short Answer

Step by step solution

Isolate the variable term

Solve for x

Key Concepts

Isolation of Variable

Inequality Reversal

Dividing by Negative Numbers

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