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Chapter 8: Problem 97

Solve each inequality. Graph the solution set and write it in interval notation. $$ 0.4 x+0.4 \leq 0.1 x+0.85 $$

Short Answer

Expert verified
The solution is \((-\infty, 1.5]\).

Step by step solution

01

Rearrange the Inequality

We start with the inequality: \[0.4x + 0.4 \leq 0.1x + 0.85\]To isolate the term with \(x\) on one side, we subtract \(0.1x\) from both sides:\[0.4x - 0.1x + 0.4 \leq 0.85\]
02

Simplify the Expression

Let's simplify the expression obtained:\[0.3x + 0.4 \leq 0.85\]Now, subtract \(0.4\) from both sides to isolate the term with \(x\):\[0.3x \leq 0.85 - 0.4\]
03

Solve for x

Calculate the difference on the right side:\[0.3x \leq 0.45\]Now, divide both sides by \(0.3\) to solve for \(x\):\[x \leq \frac{0.45}{0.3}\]
04

Calculate the Division

Compute the division on the right side:\[x \leq 1.5\]
05

Graph the Solution

On a number line, draw a solid line from negative infinity to \(1.5\), and place a closed dot on \(1.5\) because \(x\leq1.5\) includes \(1.5\) itself.
06

Write in Interval Notation

The solution set in interval notation is:\[(-\infty, 1.5]\]

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Key Concepts

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

Inequality Notation
Inequalities are a way to express the relationship between two values when they are not equal. The inequality sign provides a variety of ways to express how one value compares to another. Here are some key inequality symbols:
  • \( < \) : Less than
  • \( \leq \) : Less than or equal to
  • \( > \) : Greater than
  • \( \geq \) : Greater than or equal to
Inequality notation is crucial in mathematics to express conditions where solutions exist over a range of values. For instance, in the inequality \( x \leq 1.5 \), the \( \leq \) symbol means that \( x \) can take any value that is less than or equal to 1.5. If this inequality was in an algebra problem, such as \( 0.4x + 0.4 \leq 0.1x + 0.85 \), you follow a series of logical steps to isolate \( x \) and determine the set of all allowable values.
Interval Notation
Interval notation is a concise way of writing an interval of numbers instead of using an inequality. It provides an easy method to represent sets of solutions. Here's a breakdown of symbols in interval notation:
  • \((a, b)\): Represents all numbers between \(a\) and \(b\), not including \(a\) and \(b\).
  • \([a, b]\): Includes all numbers between \(a\) and \(b\), and includes \(a\) and \(b\) themselves.
  • \((a, b]\): Includes all numbers between \(a\) and \(b\), but not \(a\).
  • \([a, b)\): Includes all numbers between \(a\) and \(b\), but not \(b\).
For inequalities that extend indefinitely, the symbols \(-\infty\) and \(+\infty\) are used, which cannot be reached or included. So, if the solution for an inequality is \( x \leq 1.5 \), its interval notation becomes \((-\infty, 1.5]\), indicating all numbers less than or equal to 1.5.
Graphing Inequalities
Graphing inequalities on a number line helps visualize their solutions. This process involves marking specific points and shading the regions representing all possible solutions. Here's how you can graph an inequality such as \( x \leq 1.5 \):- Start by drawing a number line.- Find 1.5 on this line.- Place a closed dot on 1.5 because the inequality includes this number (due to the \( \leq \) sign).- Shade the line to the left of 1.5 to represent all numbers less than 1.5.- The shaded region along with the closed dot signifies the solution set.By viewing a graph, you can instantly see which values satisfy the inequality, providing a powerful visual tool in problem-solving.

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