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Chapter 2: Problem 49

Solve each inequality and graph the solution on the number line. 8+x > 7

Short Answer

Expert verified
The solution to the inequality is \(x > -1\). This is represented on the number line with a shading to the right from -1.

Step by step solution

01

Simplify the inequality

In order to solve the inequality, \(8+x > 7\), we want to isolate \(x\) to one side. Thus, we subtract 8 from both sides of the inequality which gives us: \(x > 7 - 8\)
02

Calculate the value of the x

Subtracting 7 and 8, gives us \(x > -1\)
03

Represent the solution on the number line

On the number line, mark a point at -1 and then shade the area to the right of -1 since \(x\) is greater than -1. The point at -1 should not be filled in, to indicate that -1 is not included in the solution.

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Key Concepts

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

Solving Inequalities
To solve an inequality like the one given in our exercise, follow a straightforward path similar to solving equations, but with a few key differences. The exercise provides us with the inequality \(8 + x > 7\). Here鈥檚 how you can approach solving it:
  • First, isolate the variable on one side. For this, you may need to perform operations such as addition, subtraction, multiplication, or division.
  • In our specific exercise, this means subtracting 8 from both sides of the inequality.
  • This step isolates \(x\), resulting in the simplified inequality \(x > -1\).
It is important to remember that, when multiplying or dividing both sides of an inequality by a negative number, the inequality sign flips. However, in this exercise, we did not need to apply that rule.
Graphing Inequalities
Once you have solved the inequality, the next step is to graph the solution. Graphing inequalities is a helpful way to visually understand the range of possible solutions. In our case, with \(x > -1\):
  • Firstly, you identify the critical number, which is -1 in this example.
  • Instead of plotting a single dot, you will show the range that \(x\) can take on a number line.
  • The graph of \(x > -1\) will have an open circle at -1, indicating that \(-1\) itself is not included.
  • Shading to the right side of -1 shows all numbers greater than -1.
This visual representation aids in understanding the extent of the solution set.
Number Line Representation
The number line is a powerful visual tool to represent the solutions of inequalities. It allows for an intuitive understanding of where the solutions lie compared to a single number or range of numbers.
Consider the solved inequality \(x > -1\). Here is how you can present this solution on a number line:
  • Draw a horizontal line and mark numbers on it as reference points (like -3, -2, -1, 0, etc.).
  • At the critical point \(-1\), draw an open circle. This circle indicates that \(-1\) is not part of the solution.
  • Shade or draw a line extending to the right from \(-1\) to demonstrate that all numbers greater than \(-1\) are the solutions.
Using these straightforward steps, you can always translate the mathematical solution of an inequality into a clear graphical representation.

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