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Chapter 2: Problem 495

Translate and solve. Then write the solution in interval notation and graph on the number line. Ten times \(y\) is at most -110 .

Short Answer

Expert verified
The solution is \((-\infty, -11]\).

Step by step solution

01

Identify the inequality

The phrase 'Ten times y is at most -110' suggests the inequality: \[10y \leq -110\]
02

Isolate the variable y

To isolate y, divide both sides of the inequality by 10:\[\frac{10y}{10} \leq \frac{-110}{10} \y \leq -11\]
03

Write the solution in interval notation

The solution \(y \leq -11\) in interval notation is:\((-\infty, -11]\)
04

Graph the solution on a number line

Draw a number line and shade all the values to the left of -11, including -11. Indicate -11 with a solid dot since y can be equal to -11.

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Key Concepts

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

Inequality Translation
When solving word problems in mathematics, translating a sentence into a mathematical inequality is often the first step. This involves recognizing key phrases:
  • 'At most' or 'no more than' means \( \leq \ \)
  • 'At least' or 'no less than' means \( \geq \ \)
  • 'Less than' is \( \< \ \)
  • 'More than' is \( \> \ \)
In the exercise, 'Ten times y is at most -110' translates to an inequality because 'at most' signifies that the value can be equal to or less than \( -110 \). Thus, we get \[ 10y \leq -110 \].
Always take note of the language used in word problems to correctly set up your inequality.
Isolating Variables
Isolating the variable is a crucial step in solving inequalities. It allows us to simplify the inequality and find the range of possible values for the variable.
  • Start by performing the same operation on both sides of the inequality.
  • Remember, if you multiply or divide by a negative number, the direction of the inequality sign changes. This isn't needed in this particular exercise since we are dividing by a positive number.
For the given exercise, to isolate y in the inequality \( 10y \leq -110 \):
\begin{align*} \frac{10y}{10} & \leq \frac{-110}{10} \ y & \leq -11 \end{align*}
By dividing both sides by 10, we successfully isolate y, finding that y is less than or equal to \( -11 \).
Interval Notation
Interval notation is a way of writing subsets of the real number line. Unlike regular expressions, interval notation provides a cleaner and more visual representation.
In interval notation:
  • Brackets \( [ ] \) are used when the endpoints are included (inclusive).
  • Parentheses \( ( ) \) are used when the endpoints are not included (exclusive).
For the inequality \( y \leq -11 \):
  • \textbf{Left endpoint:} Since y can take any value less than or equal to \( -11 \), the left endpoint is \( -\text{infinity} \), represented as \( -\text{infinity} \).\br>
  • \textbf{Right endpoint:} Since \( -11 \) is included in the solution, we use a bracket \( [ \).
So, the interval notation for \( y \leq -11 \) is: \( (-\text{infinity}, -11] \).
This tells us all values less than or equal to \( -11 \) are part of the solution.
Number Line Graphing
Graphing on a number line is a helpful way to visualize the range of values that satisfy an inequality. This can make it easier to understand and verify your solutions.
To graph \( y \leq -11 \) on a number line, follow these steps:
  • Draw a horizontal line and mark points to represent numbers.
  • Locate and mark the significant point, \( -11 \).
  • Since \( y \leq -11 \), shade the line to the left of \( -11 \) which indicates all numbers less than \( -11 \).
  • Include a solid dot at \( -11 \) to show that \( -11 \) itself is part of the solution.
This solid dot and shading appropriately represent all numbers that are less than or equal to \( -11 \). By visualizing the inequality, it becomes clearer which values satisfy the condition.

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