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Chapter 6: Problem 38

Solve each inequality and graph the solution set on a number line. \(3 x-8<2 x+11\)

Short Answer

Expert verified
The solution set for the inequality is \(x < 19\), represented on the number line as an open circle at 19 with an arrow pointing to the left.

Step by step solution

01

Simplify the inequality

Start by getting all terms with the variable \(x\) on the same side of the inequality. Subtract \(2x\) from both sides of the equation. This gets \(x - 8 < 11\).
02

Isolate the variable

To isolate \(x\), add 8 to both sides of the inequality: \(x < 19\).
03

Graph the solution

With the solution \(x < 19\), a number line with point 19 is drawn. As \(x\) is less than 19, an open circle (because 19 is not included in the solution set) at 19 is used and an arrow point to the left (towards the smaller numbers) from this point is drawn on the number line to represent the values of x satisfying the inequality.

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Key Concepts

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

Solving Linear Inequalities
Solving linear inequalities is quite similar to solving linear equations. However, the main goal when solving inequalities is to find the range of possible values that a variable can take to make the inequality true. Here's a step-by-step guide to solving the inequality \(3x - 8 < 2x + 11\):
  • First, rearrange the inequality so that all terms involving the variable are on one side. For example, in the inequality \(3x - 8 < 2x + 11\), subtract \(2x\) from both sides to get \(x - 8 < 11\).
  • Next, bring the constant terms to the opposite side to isolate the variable. Add 8 to both sides resulting in \(x < 19\).
  • Remember that performing similar operations, like adding or subtracting the same number from both sides, does not change the inequality sign.
This results in a solution where \(x\) can be any value less than 19. It's crucial to note that specific operations, such as multiplying or dividing by a negative number, require reversing the inequality sign. However, in this case, it is not needed.
Graphing Solutions
Once you solve the inequality and find the range of possible values, the next step is to translate this solution onto a graph. Graphing provides a visual understanding of the solution set. Here's how to effectively graph the solution \(x < 19\) on a number line.
  • Begin by drawing a simple horizontal line which will represent the number line.
  • Mark the number 19 on this number line to identify a crucial point in your solution.
  • Since the inequality \(x < 19\) means that 19 itself is not included in the solution, use an open circle to indicate this point on the line.
  • Finally, draw an arrow starting from the circle at 19 that extends to the left, signifying all real numbers less than 19 are part of the solution set.
The open circle and arrow clarify the solution range, providing an immediate visual cue of where the variable \(x\) lies. Understanding how to graph these solutions helps you visually interpret inequalities effectively, much like with equations but with an emphasis on ranges.
Number Line Representation
The number line is a powerful tool for representing inequalities because it clearly displays the range of values that satisfy the inequality condition. Using the number line allows for an easily understandable illustration of solutions.
  • When graphing \(x < 19\), it's important to place an open circle at the number 19. This indicates that while \(x\) can approach 19, it cannot actually be 19.
  • The arrow pointing to the left from the open circle signifies that all values to the left (or less than 19) are included in the solution.
  • For inequalities that include the limit (such as \(x \leq 19\)), a closed circle would be used to show that 19 is included.
This representation eases comprehension of inequalities and reinforces the concept of solution sets involving ranges. Recognizing these visual elements such as open and closed circles, and the direction of the arrow, simplifies interpreting the possible values that satisfy the inequality.

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Most popular questions from this chapter

Solve each equation by the method of your choice. \(x(x+1)=4-(x+2)(x+2)\)

Solve the quadratic equations in Exercises 37-52 by factoring. \(3 x^{2}-4 x=15\)

Explain how to solve a quadratic equation by factoring. Use the equation \(x^{2}+6 x+8=0\) in your explanation.

Use FOIL to find the products in Exercises 1-8. \((x-5)(x+3)\)

In an inequality such as \(5 x+4<8 x-5\), I can avoid division by a negative number depending on which side \(I\) collect the variable terms and on which side I collect the constant terms.
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