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Chapter 3: Problem 30

Solve each inequality. $$7(y+1)-(6 y+7) \leq 4$$

Short Answer

Expert verified
y \leq 4

Step by step solution

01

- Distribute and Simplify

Start by distributing the 7 through the parentheses in the expression on the left-hand side:\[7(y+1) - (6y + 7) \ = 7y + 7 - 6y - 7\]Combining like terms, this simplifies to:\[y + 0 \ = y\]So the inequality becomes:\[y \ \ \ \leq\ 4\resizebox*{!}{\baselineskip}{\sqrt{2}}4 \ \aboutsolvesome\flatdash stealthdecryptedit refining folding reducing identifying maximizing sharpening trimming a dual challengewise in lining}\]
02

- Solve for y

The simplified inequality is now:\[y \leq 4\]No further steps are needed.

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Key Concepts

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

distributive property
The distributive property is a crucial concept used widely in algebra that helps in simplifying expressions involving parentheses. It is represented as: \[a(b + c) = ab + ac\]In this exercise, we applied the distributive property to the term \[7(y+1)\], breaking it down into smaller parts. Here's the detailed breakdown:
  • \[7(y+1) = 7 \times y + 7 \times 1\] becomes \[7y + 7\]
Using the distributive property makes it easier to eliminate the parentheses and simplify the equation. Remember, the goal is to isolate the variable and solving inequalities often starts with simplifying complex expressions using the distributive property.
combining like terms
Combining like terms is another fundamental concept in algebra that lets us simplify expressions by merging terms that have the same variable raised to the same power. In this exercise, we had the expression: \[7y + 7 - 6y - 7\]Let's break it down:
  • Combine the \'y\' terms: \[7y - 6y\]
  • Combine the constants: \[7 - 7\]
So, \[7y - 6y\] simplifies to \[y\].And, \[7 - 7\] cancels out to \[0\].These steps help in reducing the problem to a simpler form, making it easier to solve for the variable.
linear inequalities
Linear inequalities are expressions where one side is not strictly equal to the other but is instead either less than, greater than, less than or equal to, or greater than or equal to the other side. In this exercise, our final inequality is: \[y \leq 4\]To solve linear inequalities:
  • Simplify both sides if necessary (using distribution, combining like terms, etc.)
  • Isolate the variable on one side of the inequality symbol
  • Remember that if you need to multiply or divide by a negative number, you must reverse the inequality sign
These steps ensure that the inequality is solved correctly. The solution \[y \leq 4\] tells us that the value of \[y\] can be 4 or any number less than 4, which represents all possible solutions to the inequality.

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Most popular questions from this chapter

Solve the inequality and sketch the solution set on a number line. $$-3 \leq y-5 \leq 4$$

Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. As dry air rises, it expands owing to the lower atmospheric pressure, and as a result the air cools at the rate of approximately \(5.4^{\circ} \mathrm{F}\) for each \(1,000\) feet of increase in altitude (up to an altitude of approximately \(40,000 \mathrm{ft}\) ). Thus, if the ground-level temperature is \(46^{\circ} \mathrm{F}\), the temperature at an altitude of \(A\) feet above the ground is given by the equation $$ T=46-0.0054 A $$ (a) Determine the temperature at an altitude of \(5,000\) feet. (b) Determine the altitude at which the temperature will be \(32^{\circ} \mathrm{F}\).

Separate the following equations into groups of equivalent equation \(=\) \(\begin{aligned} x+1 &=16 \\ x &=-1 \\ x+7 &=6 \end{aligned}\) \(\begin{aligned} 2 x &=18 \\ x-3 &=12 \\ x &=9 \end{aligned}\) \(\begin{aligned} 2 x &=10 \\ x+2 &=11 \\ 5 x &=25 \end{aligned}\) \(\begin{aligned} x &=15 \\ 2 x &=30 \\ x-2 &=7 \end{aligned}\)

Solve the inequality and sketch the solution set on a number line. $$3(a+2)-5 a \geq 2-a$$

Solve the given inequality. Round off your answers to the nearest hundredth where necessary. $$0.5(x-0.3)>0.25(x+3)$$
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