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Chapter 8: Problem 29

Graph the solution set, and write it using interval notation. $$ 3 k+1<-20 $$

Short Answer

Expert verified
The solution is (- 鈭�, -7).

Step by step solution

01

- Subtract 1 from both sides

Start by subtracting 1 from both sides of the inequality to simplify it. This gives: 3k + 1 - 1 < -20 - 1
02

- Simplify the inequality

Simplify both sides: 3k < -21
03

- Divide both sides by 3

Divide both sides of the inequality by 3 to solve for k: k < -7
04

- Write the solution in interval notation

The solution represents all numbers less than -7. In interval notation, this is written as: (- 鈭�, -7)
05

- Graph the solution

To graph the solution on a number line, draw an open circle at -7 and shade to the left, indicating all numbers less than -7.

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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 method used to represent a range of values on a number line. It provides a clear and concise way to express solutions for inequalities.

Intervals are written within brackets or parentheses:
  • Brackets \[ ..., ... \] include the endpoint values.
  • Parentheses ( ... , ... ) exclude the endpoint values.
For example:
  • (-鈭�, -7) means all values less than -7, but not including -7.
  • [-7, 5] means all values from -7 to 5, including both -7 and 5.
In our exercise, the solution of the inequality was k < -7. This means the interval notation is (-鈭�, -7).
Solving Inequalities
Solving inequalities is a process similar to solving equations. The key difference is that you must consider the direction of the inequality sign (<, >, 鈮�, 鈮�). Here's how to approach solving inequalities:
  • Step 1: Isolate the variable on one side of the inequality. You can add, subtract, multiply, or divide both sides. Be careful when multiplying or dividing by a negative number; this reverses the inequality sign.
  • Step 2: Simplify the expression if necessary. Combine like terms and reduce fractions.
  • Step 3: Express the solution in a clear form, such as interval notation or a number line.
Applying these steps to our exercise:
  • Step 1: Subtract 1 from both sides: \[3k + 1 - 1 < -20 - 1 \]
  • Step 2: Simplify: \[3k < -21 \]
  • Step 3: Divide both sides by 3: \[k < -7 \]
The result is k < -7.
Number Line Graphing
Graphing inequalities on a number line is a helpful visual tool. It shows the range of possible solutions at a glance. Here鈥檚 how you can graph inequalities:
  • Identify the critical point or boundary of the inequality. For our case, it's -7.
  • Draw an open circle around -7. This indicates that -7 is not included in the solution.
  • Shade to the left of -7 to represent all values less than -7.
The final number line graph visually represents our interval: (-鈭�, -7). Always ensure that the number line extends far enough in both directions to accurately represent the full range of potential solutions.

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Most popular questions from this chapter

Graph the solution set, and write it using interval notation. $$ -8 \leq-4 x+2 \leq 6 $$

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Graph the solution set of each system of linear inequalities. $$ \begin{array}{l} y \leq 2 x-5 \\ x<3 y+2 \end{array} $$

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Graph the solution set, and write it using interval notation. $$ 4 \leq-9 x+5<8 $$
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