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Chapter 1: Problem 119

solve and graph the solution set on a number line. $$\frac{x+3}{4} \geq \frac{x-2}{3}+1$$

Short Answer

Expert verified
The solution to the given inequality is \(x \leq 5\). The graph on a number line involves a solid circle at 5 and all numbers less than 5 being darkened.

Step by step solution

01

Clear Out the Fractions

To make the inequality easier to deal with, a good first step is to clear out the fractions by finding a common multiple of the denominators 4 and 3, which is 12. Multiply each term in the equation by 12 to achieve this: \(12 \cdot \frac{x+3}{4} \geq 12 \cdot (\frac{x-2}{3}+1)\). This simplifies to \(3(x+3) \geq 4(x-2) + 12\) or \(3x+9 \geq 4x-8+12\).
02

Combine Like Terms

Now, simplify each side of the inequality by combining like terms: \(3x+9 \geq 4x+4\).
03

Isolate Variable x

Get all x terms to one side and constant terms to the other side: subtract 3x from both sides to have \(9 \geq x+4\), then subtract 4 from both sides to find \(x \leq 5\).
04

Graph x on a Number Line

Place a solid circle at 5 on a number line to signify that the solution includes 5. Because the function is less than or equal to 5, darken all numbers less than 5 on the number line.

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Key Concepts

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

Number line
The number line is a straight, horizontal line that helps us visually represent real numbers. It extends infinitely in both directions, with zero positioned at the center. Positive numbers appear to the right of zero, while negative numbers populate the left. Using a number line makes it easy to visualize solutions to inequalities. When graphing solutions on a number line, we often use different symbols to depict the range of numbers included in the solution:
  • A solid circle is used when the number itself is part of the solution, such as when we have "greater than or equal to" (鈮�) or "less than or equal to" (鈮�).
  • An open circle represents a number not included in the solution, typical for "greater than" (>) or "less than" (<) inequalities.
Imagine plotting points along the number line as part of a treasure map, indicating where you can and cannot go. It provides a quick visual reference for students and is a vital tool for understanding inequalities.
Solving inequalities
If you've solved equations before, solving inequalities will feel quite similar. However, instead of finding an exact number, you'll find a range of possible values for the variable that make the inequality true. The steps we follow for solving inequalities often involve:
  • Eliminating Fractions: Clear fractions by multiplying all terms by the least common multiple of the denominators. This helps simplify the inequality for easier manipulation.
  • Combining Like Terms: Like in equations, organize and simplify the inequality by bringing similar terms together.
  • Isolating the Variable: Get the variable of interest on one side of the inequality to see the range it's allowed to take.
Remember, when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be flipped. This is a unique aspect of inequalities that helps maintain the truth of the statement.
Graphical representation of inequalities
Visualizing inequalities through graphs on a number line can clarify the nature of the solutions we discover. Let's take the inequality solution from the exercise, which concluded with \(x \leq 5\).On the number line, we begin by marking the point 5:
  • We place a solid circle on the number 5, indicating that 5 is included in the solution set.
  • Then, we shade to the left of 5, covering all numbers smaller than 5, showing all potential values of \(x\) that satisfy the inequality.
This visual method not only strengthens our understanding of inequalities but also translates abstract math concepts into a clear, mental image. Such representations support answers' validity and help cross-check the algebraic work with an intuitive, pictorial form.

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