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Chapter 0: Problem 44

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$7-\frac{4}{5} x<\frac{3}{5}$$

Short Answer

Expert verified
The solution to the inequality is \(x <8\). In interval notation, this is \(-\infty , 8)\).

Step by step solution

01

Isolate the variable

The aim is to isolate \(x\) in the inequality. Start by subtracting 7 from both sides of the equation: \(-\frac{4}{5}x <\frac{3}{5}-7\). After the subtraction, the equation reduces to \(-\frac{4}{5}x <-\frac{32}{5}\).
02

Solve for x

To solve for \(x\), divide the inequality by -\(\frac{4}{5}\). Remember when dividing or multiplying an inequality by a negative number, the direction of the inequality must switch, it becomes \(\frac{-\frac{32}{5}}{-\frac{4}{5}}>x\), which simplifies to \(x < 8\).
03

Write the solution in interval notation

The solution in interval notation is \(-\infty , 8)\), which means that all values less than 8 are valid solutions.
04

Graph the solution set

On a number line, point at number 8 is an open dot, indicating that 8 itself isn't included in the solution set, and all values to the left of 8 are shaded, demonstrating that they are part of the solution set.

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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 concise way of expressing a range of numbers on a number line. It helps to indicate which numbers are included or not in a solution set. In our exercise, the solution to the inequality is expressed as \((-\infty, 8)\).

Here's how interval notation works:
  • Parentheses \(()\) are used to show that an endpoint is not included, known as an "open" interval.
  • Brackets \([]\) indicate that an endpoint is included, known as a "closed" interval.
For \((-\infty, 8)\), the infinity symbol \(-\infty\) always has a parenthesis since infinity isn't a number we can reach. The parenthesis at 8 means 8 itself isn't included in the solutions.
Solving Inequalities
Solving inequalities involves finding all values of a variable that make the inequality true. Here's how you can solve inequalities such as the one in our example, \(7-\frac{4}{5}x<\frac{3}{5}\):

  • Isolate the Variable: Get the variable on one side of the inequality. In this problem, we subtracted 7 from both sides resulting in \(-\frac{4}{5}x < -\frac{32}{5}\).
  • Multiply or Divide: To solve for \(x\), divide both sides by \(-\frac{4}{5}\), remembering to flip the inequality sign when dividing by a negative. This step yields \(x < 8\).
Understanding when to flip the inequality sign is crucial, as it can change the entire solution if not done correctly.
Number Line Graphing
Graphing solutions on a number line provides a visual representation of the solution set. Let's see how this applies to our inequality solution of \(x < 8\):

  • Identify the Endpoint: Place an open dot at the number 8 on the number line. This shows that 8 is not included in the solution set.
  • Shade the Solution Region: Color the line to the left of 8, indicating that all numbers less than 8 are solutions.
This graph clearly communicates the values that satisfy the inequality, making it easier to visualize the solution set.

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is \(\$ 3000\) and it costs \(\$ 3,00\) to produce cach package of stationery. The selling price is \(\$ 5.50\) per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?

Perform the indicated operations. $$(x-y)^{-1}+(x-y)^{-2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\).

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I want to solve \(25 x^{2}-169=0\) fairly quickly, I'll use the quadratic formula.

Explain how to find the least common denominator for denominators of \(x^{2}-100\) and \(x^{2}-20 x+100\).
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