/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 73 Solve each inequality. \(\frac... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 73

Solve each inequality. \(\frac{y-1}{15}>-\frac{2}{5}\)

Short Answer

Expert verified
The solution is \( y > -5 \).

Step by step solution

01

Clear the Fraction

To eliminate the fraction on the left-hand side, multiply every term by 15. This is done to remove the denominator.\[ 15 \times \frac{y - 1}{15} > 15 \times \left( -\frac{2}{5} \right) \]This yields:\[ y - 1 > -6 \]
02

Isolate the Variable

Add 1 to both sides of the inequality to solve for \( y \).\[ y - 1 + 1 > -6 + 1 \]Which simplifies to:\[ y > -5 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Fractions in algebra
Fractions in algebra can often seem daunting, but simplifying them is a crucial skill. When working with fractions, think about them as divisions; the numerator is divided by the denominator.
In equations and inequalities, we aim to eliminate fractions to create simpler expressions. This process often involves multiplying all terms by the denominator.
  • For example, with the inequality \(\frac{y-1}{15}>-\frac{2}{5}\), we multiplied both sides by 15.
  • Multiplying by 15 cancels out the denominator in the fraction on the left side, simplifying the inequality.
Clearing fractions in this way helps streamline solving processes, reducing the equation or inequality to a more familiar form with whole numbers. This conversion is a stepping stone to solving algebraic equations more comfortably and efficiently.
Isolating variables
Isolating the variable is a fundamental step in solving equations and inequalities. The term "isolating the variable" refers to rearranging the equation so that the variable appears by itself on one side of the equation or inequality.
To isolate the variable, you generally perform inverse operations to move other terms away from the variable.
  • In our inequality \( y - 1 > -6 \), we added 1 to both sides.
  • Adding 1 is the inverse of subtracting 1, effectively "undoing" the subtraction and isolating \(y\).
This isolation clears the path to finding the solution. Thinking of it piece-by-piece, moving each component systematically, helps in maintaining the balance of the equation or inequality.
Inequality solutions
Solving an inequality is similar to solving an equation, but with some essential differences. The solution to an inequality is a range of values, not just a single number.
In this example, the original inequality \(\frac{y-1}{15}>-\frac{2}{5}\) turned into \(y > -5\) after clearing fractions and isolating the variable.
  • This means any value of \(y\) greater than \(-5\) satisfies the inequality.
  • It's crucial to note that, unlike equations, inequalities can have a range of solutions rather than a specific single value.
When solving inequalities, always remember to consider directionality; if you multiply or divide both sides of an inequality by a negative number, the inequality sign flips direction. This ensures that you correctly interpret the range of valid solutions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. See Examples 6 and 7 . $$ f(x)=-\frac{1}{4} x^{2} $$

Sketch the graph of each quadratic function. Label the vertex, and sketch and label the axis of symmetry. $$ f(x)=\frac{1}{4} x^{2}-9 $$

Solve by completing the square. See Section 11.1. $$ y^{2}+6 y=-5 $$

Use the quadratic formula to solve each quadratic equation. \(3 x^{2}-\sqrt{12} x+1=0\) (Hint: \(a=3, b=-\sqrt{12}, c=1)\)

Use the quadratic formula and a calculator to approximate each solution to the nearest tenth. $$ 2 x^{2}-6 x+3=0 $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.