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Chapter 8: Problem 20

Graph the solution set, and write it using interval notation. $$ -\frac{2}{3} x \leq 12 $$

Short Answer

Expert verified
The solution set is \( [-18, \infty) \) and it is represented on the number line with a closed dot at -18 and shading to the right.

Step by step solution

01

- Isolate the variable

First, isolate the variable by multiplying both sides of the inequality by the reciprocal of -\(\frac{2}{3}\), which is -\(\frac{3}{2}\). This step includes reversing the inequality sign because we are multiplying by a negative number. \[ -x \leq 12 \cdot \left(-\frac{3}{2}\right) \]The inequality becomes: \[ x \geq -18 \]
02

- Write in interval notation

The solution \( x \geq -18 \) can be written in interval notation. This means the variable \( x \) includes all numbers greater than or equal to -18. The interval notation is: \( [-18, \infty) \)
03

- Graph the solution

Graph the solution on the number line. First, draw a number line and mark the point -18 with a closed dot to indicate that it is included in the solution. Then, shade the region to the right of -18 to represent all numbers greater than or equal to -18.

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Key Concepts

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

Interval Notation
Interval notation is a way to represent the set of solutions for an inequality or a range of values. It uses brackets and parentheses to describe the range. For example, if we have the solution set for the inequality \(x \geq -18\), it means that x can be any number starting from -18 up to infinity.
  • An open parenthesis \(()\) signifies that the endpoint is not included.
  • A closed bracket \([\) signifies that the endpoint is included.
Therefore, \[x \geq -18\] would be written in interval notation as \([-18, \infty)\). Here, we use a closed bracket around -18 because x can be equal to -18, and an open parenthesis around \infty because infinity is not a number we can actually equal.
Practice writing and recognizing interval notations to make working with solution sets easier.
Number Line Graphing
Graphing solutions on a number line helps visualize the range of values that satisfy an inequality. Here鈥檚 how to graph the solution \(x \geq -18\):
  • Draw a horizontal line and select a point to represent -18.
  • Place a closed dot at -18 to indicate that -18 is included in the solution set.
  • Shade the entire region to the right of -18 to show that all numbers greater than -18 are also part of the solution set.
The graph provides a clear visual representation of where the solution set lies on the number line. Remember that a closed dot means the point is included (indicated by a \[\]), while an open dot means it is not (indicated by a \(\)).
This method is especially helpful for understanding more complex inequalities and seeing how different inequalities relate to each other.
Multiplying Inequalities by Negative Numbers
One important rule in solving inequalities is that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.
  • For example, if you have \(-x \leq 12\), multiplying both sides by -1 will change it to \(x \geq -12\).
  • The reason behind this rule is that the order of numbers reverses when multiplied or divided by a negative.

In the given problem, \(-\frac{2}{3}x \leq 12\), multiplying both sides by \(-\frac{3}{2}\) (the reciprocal) both isolates the variable and reverses the inequality. This ensures we can find the true range of x values that solve the inequality. Pay attention to this rule, as it is crucial for solving inequalities correctly.
Understanding why we flip the inequality can help remember this step and avoid mistakes in your solutions.

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Most popular questions from this chapter

Concept Check Express each set in simplest interval form. [-1,2]\(\cup(0,5)\)

Graph the solution set of each system of linear inequalities. $$ \begin{array}{r} 3 x+y>4 \\ x+2 y<2 \end{array} $$

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Solve each compound inequality. Graph the solution set, and write it using interval notation. \(x+1 \geq 5\) and \(x-2 \leq 10\)

Graph the solution set of each system of linear inequalities. $$ \begin{aligned} x-3 y & \leq 6 \\ x & \geq-5 \end{aligned} $$
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