/*! 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 \text { Show a step-by-step symb... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 11

\text { Show a step-by-step symbolic solution of the inequality }-3 x+4>16

Short Answer

Expert verified
The solution is \(x < -4\).

Step by step solution

01

Subtract 4 from Both Sides

Start by isolating the term with the variable. Subtract 4 from both sides of the inequality:\[-3x + 4 - 4 > 16 - 4\]This simplifies to:\[-3x > 12\]
02

Divide by -3 and Reverse Inequality

To solve for \(x\), divide both sides by \(-3\). Remember to reverse the inequality sign because we are dividing by a negative number:\[x < \frac{12}{-3}\]\[x < -4\]

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.

Understanding Algebra
Algebra is like a puzzle that uses symbols and numbers to solve problems. It allows us to represent real-world situations using letters in place of unknown numbers or values.
You'll often encounter letters like \(x\) or \(y\) standing in for numbers we don't yet know. This is helpful because it lets us write and solve equations to find out what those numbers are.
  • In the example above, \(-3x + 4 > 16\), \(x\) represents a number we're trying to find.
  • Algebra involves combining what we know to see what we can find out, working logically step by step.
Using algebraic methods, we work to isolate the variable (such as \(x\)) on one side of the equation or inequality, to find its value or range.
What is a Symbolic Solution?
A symbolic solution is a solution method that involves manipulating symbols in equations or inequalities to solve for unknown variables.
This approach relies on logical operations and known algebraic properties to reach a conclusion.
  • For the inequality \(-3x + 4 > 16\), we move terms around to get \(-3x\) by itself on one side.
  • This involves subtracting and then dividing by constants to simplify the inequality.
Symbolically solving expressions helps us understand exactly what operations are needed and avoid numerical miscalculations, as we use symbols until the very last steps.
Inequalities Reversal Rule
The inequalities reversal rule is a crucial concept when dealing with inequalities, particularly when you are multiplying or dividing both sides by a negative number.
This rule states that when you multiply or divide both sides of an inequality by a negative number, you must reverse or flip the inequality sign.
  • In step 2 of the solution, to isolate \(x\), we divided by \(-3\), which required us to reverse the sign, changing \(>\) to \(<\).
  • This is because the direction of the inequality depends on the way numbers are ordered; multiplying or dividing by a negative number flips this order.
By keeping this rule in mind, you ensure that the inequality remains true throughout the solving process. This step is essential in ensuring your final solution accurately reflects the relationships between the variables and constants in the inequality.

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

A box is made so that its length is \(6 \mathrm{~cm}\) more than its width. Its height is \(2 \mathrm{~cm}\) less than its width. a. Use the width as the independent variable, and write an equation for the volume of the box. (a) b. Suppose you want to ensure that the volume of the box is greater than \(47 \mathrm{~cm}^{3}\). Use a graph and a table to describe all possible widths, to the nearest \(0.1 \mathrm{~cm}\), of such boxes.

Consider the equation \(y=(x+1)(x-3)\). a. How many \(x\)-intercepts does the graph have? b. Find the vertex of this parabola. c. Write the equation in vertex form. Describe the transformations of the parent function, \(y=x^{2}\).

Kayleigh says that the roots of \(0=x^{2}+16\) are The roots of 4 and \(-4\) because \((4)^{2}=16\) and \((-4)^{2}=16\). \(0=x^{2}+16\) are 4 and \(-4\) Derek tells Kayleigh that there are no roots for this equation. Who is correct and why?

Rewrite each quadratic equation in general form if necessary. For each equation, identify the values of \(a, b\), and \(c\). a. \(2 x^{2}+3 x-7=0\) b. \(x^{2}+6 x=-11\) (a) c. \(-3 x^{2}-4 x+12=0\) d. \(18-4.9 x^{2}+47 x=0\) (a) e. \(-16 x^{2}+28 x+10=57\)

Draw a rectangle diagram to represent each expression. Then write an equation showing the product of the two binomials and the equivalent polynomial in general form. a. \((x+2)(x+4)\) b. \((x+3)(x+5)\) c. \((x+2)(x-5)\) d. \(x(x-3)\) (ii) e. \((x+2)(2 x+5)\) f. \((3 x-1)(2 x+3)\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

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.