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Magazine Chapter 1: Problem 46
Graph all solutions on a number line and provide the corresponding interval notation. $$ -7 \leq 3 y+5 \leq 2 $$
Short Answer
Expert verified
The solution is \([-4, -1]\).
Step by step solution
01
Understand the Compound Inequality
The inequality given is \(-7 \leq 3y + 5 \leq 2\). This is a compound inequality, meaning we need to solve for \(y\) such that both parts of the inequality are satisfied simultaneously.
02
Isolate for y in the Left Inequality
Consider the left part of the compound inequality: \(-7 \leq 3y + 5\).Subtract 5 from both sides to begin isolating \(y\):\(-7 - 5 \leq 3y \-12 \leq 3y\).
03
Solve for y in the Left Inequality
Now, divide both sides of the inequality by 3 to fully solve for \(y\):\(\frac{-12}{3} \leq \frac{3y}{3} \-4 \leq y\).
04
Isolate for y in the Right Inequality
Next, consider the right part of the inequality:\(3y + 5 \leq 2\).Subtract 5 from both sides:\(3y + 5 - 5 \leq 2 - 5 \3y \leq -3\).
05
Solve for y in the Right Inequality
Divide both sides of the inequality by 3:\(\frac{3y}{3} \leq \frac{-3}{3} \y \leq -1\).
06
Combine the Solutions
From the steps above, we have two inequalities:\(-4 \leq y\) and \(y \leq -1\).Combine them to form the solution for the compound inequality:\(-4 \leq y \leq -1\).
07
Graph on a Number Line
Graph the solution set \(-4 \leq y \leq -1\) on a number line by drawing a closed circle at \(-4\) and \(-1\), then shading the region in between.
08
Interval Notation
Express the solution set in interval notation. Since both endpoints are included (closed circles), use square brackets:\([-4, -1]\).
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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 mathematical notation used to describe a range of values. It simplifies expressing the set of all numbers between two endpoints. For example, when a compound inequality like \(-4 \leq y \leq -1\) obtains, we describe this solution in interval notation as \([-4, -1]\). In this notation:
- Square brackets \([]\) are used when the endpoints are included in the solution, indicating a closed interval.
- Parentheses \(()\) signify that the endpoints are not included, indicating an open interval.
Number Line Graphing
Graphing on a number line involves representing the set of solutions visually, which helps in understanding the range of values that solve the inequality. For the compound inequality \(-4 \leq y \leq -1\), graphing involves:
- Drawing a horizontal line to serve as the number line with equal spaced increments.
- Placing closed circles on the endpoint values \(-4\) and \(-1\) to indicate that these values are included in the solution set.
- Shading the region between these closed circles to show all the possible values of \y\ within the interval.
Solving Inequalities
To solve inequalities, especially compound inequalities like \(-7 \leq 3y + 5 \leq 2\), we break down the process in manageable steps. The primary goal is to isolate the variable, often \y\, to find which values balance both the inequalities.
- Start by isolating \y\ in both inequalities separately. For example: \-7 \leq 3y + 5\ and \3y + 5 \leq 2\.
- Subtract or add constants to both sides to simplify, such as subtracting 5 in both parts, giving \-12 \leq 3y \ and \3y \leq -3\.
- Divide each part by the coefficient of \y\, which in this case is 3, to solve for \y\, leading to \-4 \leq y\ and \y \leq -1\.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations which collectively form parts of equations or inequalities. In the compound inequality \(-7 \leq 3y + 5 \leq 2\), \3y + 5\ is the algebraic expression.
- Each algebraic expression can be simplified by using basic arithmetic operations like addition, subtraction, multiplication, and division.
- They play a critical role in formulating expressions that can then be evaluated to find unknown values.
- Understanding how to manipulate and solve algebraic expressions is crucial in solving inequalities and finding ranges of solutions.
