/*! 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 11 Solve the given inequalities. Gr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 17: Problem 11

Solve the given inequalities. Graph each solution. $$12-2 y>16$$

Short Answer

Expert verified
The solution is \(y < -2\), and it's graphed with an open circle at \(-2\) and shading to the left.

Step by step solution

01

Isolate the variable term

Start by isolating the term with the variable on one side of the inequality. Given the inequality \(12 - 2y > 16\), subtract 12 from both sides to get \(-2y > 4\).
02

Solve for the variable

Now, solve for \(y\) by dividing both sides by \(-2\). Remember that when you divide or multiply an inequality by a negative number, the inequality sign flips. Thus, \(-2y > 4\) becomes \(y < -2\).
03

Graph the solution

To graph the solution \(y < -2\), draw a number line. Place an open circle at \(-2\) to indicate that \(-2\) is not included in the solution. Shade the line to the left of \(-2\) to represent all numbers less than \(-2\).

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.

Solving Inequalities
Inequalities involve expressions where two values are compared using inequality signs such as greater than (>), less than (<), greater than or equal to (鈮�), or less than or equal to (鈮�). To solve inequalities, the goal is to isolate the variable on one side while keeping the inequality true. This process is similar to solving equations, but there is one important additional rule.
  • When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This step is vital to maintain the inequality's truth.
  • For example, in the inequality given: \(12 - 2y > 16\), first subtract 12 from both sides to isolate the term with the variable, leading to \(-2y > 4\).
  • To solve for \(y\), divide both sides by \(-2\). Since you're dividing by a negative, flip the sign to obtain \(y < -2\).
Be attentive to these transformations, checking each step to ensure you maintain the inequality conditions correctly.
Inequality Graphing
Graphing inequalities allows you to vividly represent solutions by showing all possible values satisfying the inequality condition. Once you have solved an inequality and found the range of solutions, you graph these on a number line.
  • For the solution \(y < -2\), use an open circle at \(-2\). The open circle signifies that \(-2\) itself is not part of the solution set.
  • Shade to the left because the solution includes all values less than \(-2\).
  • By visually representing \(y < -2\) on the number line, you clarify which numbers meet the inequality condition.
Inequality graphing serves as a quick visual cue, perfectly depicting ranges in solutions without needing to list them all.
Number Line Representation
A number line representation of an inequality solution makes it easier to understand and visualize the set of possible values. It helps in quickly interpreting the range an inequality covers.
  • The number line should be neatly labeled with numbers, typically including at least the critical point(s) involved in your inequality.
  • For \(y < -2\), an open circle on \(-2\) symbolizes exclusion from the solution set. The openness indicates that while \(-2\) is significant, it isn't included.
  • Shading or elongating a line towards negative infinity (to the left) demonstrates the endlessness of possible values satisfying the condition \(y < -2\).
By regularly practicing number line representation, you enhance your understanding and ability to swiftly solve and interpret inequalities.

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

Solve the given inequalities. Graph each solution. $$3 x-5 \leq-11$$

Solve the given linear programming problems. An oil refinery refines types \(A\) and \(B\) of crude oil and can refine as much as 4000 barrels each week. Type A crude has 2 kg of impurities per barrel, type \(\mathrm{B}\) has \(3 \mathrm{kg}\) of impurities per barrel, and the refinery can handle no more than \(9000 \mathrm{kg}\) of these impurities each week. How much of each type should be refined in order to maximize profits, if the profit is \(\$ 4 /\) barrel for type \(\mathrm{A}\) and \$5/barrel for type B?

Solve the given inequalities. Graph each solution. $$\frac{1}{3} x+2 \geq 1$$

give verbal statements equivalent to the given inequalities involving the number \(x\). x < -10 \text { or } 10 \leq x < 20

For the inequality \(4<9,\) state the inequality that results when the given operations are performed on both members. Add 3.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.