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Chapter 7: Problem 4

Solve the given inequality graphically: $$ 5 x-3>0 $$

Short Answer

Expert verified
The solution is \(x > \frac{3}{5}\).

Step by step solution

01

Isolate the Variable

First, let's isolate the variable \(x\) in the inequality \(5x - 3 > 0\). We do this by adding 3 to both sides of the inequality, which gives us: \[5x > 3\]
02

Solve for x

Next, divide both sides of the inequality \(5x > 3\) by 5 to solve for \(x\). This will yield: \[x > \frac{3}{5}\]
03

Graph the Solution

Plot the solution on a number line. Draw an open circle at \(x = \frac{3}{5}\) because \(x = \frac{3}{5}\) is not included in the solution. Shade the region to the right of \(x = \frac{3}{5}\) to indicate that \(x\) can be any value greater than \(\frac{3}{5}\).
04

Interpret the Graph

The graph shows all the values of \(x\) that solve the inequality. The solution is all real numbers greater than \(\frac{3}{5}\).

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Key Concepts

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

Isolating the Variable
When working with inequalities like the one in our exercise, the first step is to isolate the variable. This means we want to get the variable, in this case, \(x\), by itself on one side of the inequality.
To do this with the equation \(5x - 3 > 0\), we first add 3 to both sides, which simplifies the inequality to \(5x > 3\).
The next step involves dividing each side by 5, giving us \(x > \frac{3}{5}\).
By performing these operations:

  • Addition or subtraction to remove constants from the variable side
  • Division or multiplication to solve for the variable
we effectively isolate the variable. Remember, if you multiply or divide by a negative number, the inequality sign flips. But here, since we divided by a positive number, the inequality sign remains the same.
Number Line Plot
A number line plot is a visual way to show the solution of an inequality. It helps to clearly understand which values satisfy the condition set by the inequality.
To create a number line plot for \(x > \frac{3}{5}\), you start by locating the position of \(\frac{3}{5}\) on the line.
Since \(x\) is greater than \(\frac{3}{5}\), you will shade the line to the right of this point. This shading represents all numbers that are solutions to the inequality.
Number line plots are extremely helpful to:
  • Visualize the range of possible solutions
  • Comprehend the effect of the inequality sign
  • Determine inclusion or exclusion of boundary values
Open Circle Notation
In inequalities, open circle notation is used to show that a particular boundary number is not included in the solution. An open circle is drawn over the number on the number line.
For our inequality \(x > \frac{3}{5}\), we place an open circle on \(\frac{3}{5}\). This shows that \(x\) cannot equal \(\frac{3}{5}\), but it can be any number slightly larger.
Understand that:
  • An open circle means 'not included,' suitable for \(>\) or \(<\) inequalities.
  • A closed circle means 'included,' used with \(\geq\) or \(\leq\) inequalities.
This notation makes it easy to quickly see if boundary values are part of the solution set.

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Most popular questions from this chapter

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