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Chapter 1: Problem 52

Solve the inequality. Graph the solution. $$ -3 x>24 $$

Short Answer

Expert verified
The solution to the inequality is \(x < -8\). On the number line, this would be depicted by an open circle on -8, with a line extending to the left from -8 to symbolize all numbers less than -8.

Step by step solution

01

Divide both sides by -3

The idea is to get \(x\) on one side of the inequality by itself. To do this, divide both sides by -3. But remember, when dividing an inequality by a negative number the inequality symbol reverses.So, \(-3x > 24\) would become \(\frac{-3x}{-3} < \frac{24}{-3}\).
02

Simplification

After doing the division we get \(x < -8\). So, \(x\) is less than -8.
03

Graphing the solution

On the number line, the solution \(x < -8\) can be plotted. Start by drawing a number line and marking the number -8. Here, use an open circle to represent that -8 is not part of the solution and draw a line to the left of the -8 to represent numbers less than -8.

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Key Concepts

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

Graphing Inequalities
Graphing inequalities is a useful way to visualize the set of numbers that satisfy an inequality. When graphing a solution, start by identifying if the inequality is '<', '>', '鈮�', or '鈮�'. In our exercise, we found that
  • The solution to \( -3x > 24 \) transforms into \( x < -8 \) after switching sides and dividing by a negative number.
  • To represent this on the graph, locate -8 on the number line.
  • An open circle is used at -8 to show that -8 is not included in the solution.
  • Draw a ray extending to the left, indicating all numbers less than -8 satisfy the inequality.
This visual tool helps confirm the range of solutions in a simple and effective way.
Number Line
A number line is a straight line where each point corresponds to a real number. It makes understanding the range of possible solutions clear for inequalities. On the number line:
  • Positive numbers are placed to the right, and negative numbers to the left of zero.
  • The space between each tick typically represents one unit or any consistent value.
  • Numbers increase as you move to the right and decrease as you move to the left.
To graph an inequality like \( x < -8 \), you:
  • Draw the number line and mark the point corresponding to -8.
  • Use an open circle at -8, suggesting that the value -8 itself is not included in the set of solutions.
  • Shade or draw a line extending leftwards from this point, denoting all numbers smaller than -8 are included.
Inequality Symbols
Inequality symbols are crucial in expressing relationships between two expressions. In the context of our exercise:
  • '<' means "less than," representing values below a specific number.
  • '>' means "greater than," which isn't used in our final solution but was part of the original inequality.
  • '鈮�' (less than or equal to) and '鈮�' (greater than or equal to) include the number as part of the solution.
The crucial concept here is when solving inequalities, especially involving negative numbers, the inequality symbol should flip when you multiply or divide both sides by a negative number.
Negative Numbers and Inequalities
Negative numbers introduce a unique feature when solving inequalities. When you solve inequalities, any multiplication or division by a negative number causes the inequality to reverse.
For example, in our step-by-step solution of \( -3x > 24 \):
  • We divide by -3 to isolate x, which in turn flips the '>' symbol to '<'. Thus, \( x < -8 \).
This adjustment is crucial for obtaining the correct solution set. It's a common pitfall for learners and crucial to always remember during manipulation of inequalities with negative values.

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Most popular questions from this chapter

Solve the equation. $$x-23=19$$

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Your friend is having trouble understanding the Midpoint Formula. a. Explain how to find the midpoint when given the two endpoints in your own words. b. Explain how to find the other endpoint when given one endpoint and the midpoint in your own words
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