Algebra - Solutions And Solution Sets - Pauls Online Math Notes

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• Home
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• Algebra
  • 1. Preliminaries
    • 1.1 Integer Exponents
    • 1.2 Rational Exponents
    • 1.3 Radicals
    • 1.4 Polynomials
    • 1.5 Factoring Polynomials
    • 1.6 Rational Expressions
    • 1.7 Complex Numbers
  • 2. Solving Equations and Inequalities
    • 2.1 Solutions and Solution Sets
    • 2.2 Linear Equations
    • 2.3 Applications of Linear Equations
    • 2.4 Equations With More Than One Variable
    • 2.5 Quadratic Equations - Part I
    • 2.6 Quadratic Equations - Part II
    • 2.7 Quadratic Equations : A Summary
    • 2.8 Applications of Quadratic Equations
    • 2.9 Equations Reducible to Quadratic in Form
    • 2.10 Equations with Radicals
    • 2.11 Linear Inequalities
    • 2.12 Polynomial Inequalities
    • 2.13 Rational Inequalities
    • 2.14 Absolute Value Equations
    • 2.15 Absolute Value Inequalities
  • 3. Graphing and Functions
    • 3.1 Graphing
    • 3.2 Lines
    • 3.3 Circles
    • 3.4 The Definition of a Function
    • 3.5 Graphing Functions
    • 3.6 Combining Functions
    • 3.7 Inverse Functions
  • 4. Common Graphs
    • 4.1 Lines, Circles and Piecewise Functions
    • 4.2 Parabolas
    • 4.3 Ellipses
    • 4.4 Hyperbolas
    • 4.5 Miscellaneous Functions
    • 4.6 Transformations
    • 4.7 Symmetry
    • 4.8 Rational Functions
  • 5. Polynomial Functions
    • 5.1 Dividing Polynomials
    • 5.2 Zeroes/Roots of Polynomials
    • 5.3 Graphing Polynomials
    • 5.4 Finding Zeroes of Polynomials
    • 5.5 Partial Fractions
  • 6. Exponential and Logarithm Functions
    • 6.1 Exponential Functions
    • 6.2 Logarithm Functions
    • 6.3 Solving Exponential Equations
    • 6.4 Solving Logarithm Equations
    • 6.5 Applications
  • 7. Systems of Equations
    • 7.1 Linear Systems with Two Variables
    • 7.2 Linear Systems with Three Variables
    • 7.3 Augmented Matrices
    • 7.4 More on the Augmented Matrix
    • 7.5 Nonlinear Systems
• Calculus I
  • 1. Review
    • 1.1 Functions
    • 1.2 Inverse Functions
    • 1.3 Trig Functions
    • 1.4 Solving Trig Equations
    • 1.5 Trig Equations with Calculators, Part I
    • 1.6 Trig Equations with Calculators, Part II
    • 1.7 Exponential Functions
    • 1.8 Logarithm Functions
    • 1.9 Exponential and Logarithm Equations
    • 1.10 Common Graphs
  • 2. Limits
    • 2.1 Tangent Lines and Rates of Change
    • 2.2 The Limit
    • 2.3 One-Sided Limits
    • 2.4 Limit Properties
    • 2.5 Computing Limits
    • 2.6 Infinite Limits
    • 2.7 Limits At Infinity, Part I
    • 2.8 Limits At Infinity, Part II
    • 2.9 Continuity
    • 2.10 The Definition of the Limit
  • 3. Derivatives
    • 3.1 The Definition of the Derivative
    • 3.2 Interpretation of the Derivative
    • 3.3 Differentiation Formulas
    • 3.4 Product and Quotient Rule
    • 3.5 Derivatives of Trig Functions
    • 3.6 Derivatives of Exponential and Logarithm Functions
    • 3.7 Derivatives of Inverse Trig Functions
    • 3.8 Derivatives of Hyperbolic Functions
    • 3.9 Chain Rule
    • 3.10 Implicit Differentiation
    • 3.11 Related Rates
    • 3.12 Higher Order Derivatives
    • 3.13 Logarithmic Differentiation
  • 4. Applications of Derivatives
    • 4.1 Rates of Change
    • 4.2 Critical Points
    • 4.3 Minimum and Maximum Values
    • 4.4 Finding Absolute Extrema
    • 4.5 The Shape of a Graph, Part I
    • 4.6 The Shape of a Graph, Part II
    • 4.7 The Mean Value Theorem
    • 4.8 Optimization
    • 4.9 More Optimization Problems
    • 4.10 L'Hospital's Rule and Indeterminate Forms
    • 4.11 Linear Approximations
    • 4.12 Differentials
    • 4.13 Newton's Method
    • 4.14 Business Applications
  • 5. Integrals
    • 5.1 Indefinite Integrals
    • 5.2 Computing Indefinite Integrals
    • 5.3 Substitution Rule for Indefinite Integrals
    • 5.4 More Substitution Rule
    • 5.5 Area Problem
    • 5.6 Definition of the Definite Integral
    • 5.7 Computing Definite Integrals
    • 5.8 Substitution Rule for Definite Integrals
  • 6. Applications of Integrals
    • 6.1 Average Function Value
    • 6.2 Area Between Curves
    • 6.3 Volumes of Solids of Revolution / Method of Rings
    • 6.4 Volumes of Solids of Revolution/Method of Cylinders
    • 6.5 More Volume Problems
    • 6.6 Work
  • Appendix A. Extras
    • A.1 Proof of Various Limit Properties
    • A.2 Proof of Various Derivative Properties
    • A.3 Proof of Trig Limits
    • A.4 Proofs of Derivative Applications Facts
    • A.5 Proof of Various Integral Properties
    • A.6 Area and Volume Formulas
    • A.7 Types of Infinity
    • A.8 Summation Notation
    • A.9 Constant of Integration
• Calculus II
  • 7. Integration Techniques
    • 7.1 Integration by Parts
    • 7.2 Integrals Involving Trig Functions
    • 7.3 Trig Substitutions
    • 7.4 Partial Fractions
    • 7.5 Integrals Involving Roots
    • 7.6 Integrals Involving Quadratics
    • 7.7 Integration Strategy
    • 7.8 Improper Integrals
    • 7.9 Comparison Test for Improper Integrals
    • 7.10 Approximating Definite Integrals
  • 8. Applications of Integrals
    • 8.1 Arc Length
    • 8.2 Surface Area
    • 8.3 Center of Mass
    • 8.4 Hydrostatic Pressure
    • 8.5 Probability
  • 9. Parametric Equations and Polar Coordinates
    • 9.1 Parametric Equations and Curves
    • 9.2 Tangents with Parametric Equations
    • 9.3 Area with Parametric Equations
    • 9.4 Arc Length with Parametric Equations
    • 9.5 Surface Area with Parametric Equations
    • 9.6 Polar Coordinates
    • 9.7 Tangents with Polar Coordinates
    • 9.8 Area with Polar Coordinates
    • 9.9 Arc Length with Polar Coordinates
    • 9.10 Surface Area with Polar Coordinates
    • 9.11 Arc Length and Surface Area Revisited
  • 10. Series & Sequences
    • 10.1 Sequences
    • 10.2 More on Sequences
    • 10.3 Series - The Basics
    • 10.4 Convergence/Divergence of Series
    • 10.5 Special Series
    • 10.6 Integral Test
    • 10.7 Comparison Test/Limit Comparison Test
    • 10.8 Alternating Series Test
    • 10.9 Absolute Convergence
    • 10.10 Ratio Test
    • 10.11 Root Test
    • 10.12 Strategy for Series
    • 10.13 Estimating the Value of a Series
    • 10.14 Power Series
    • 10.15 Power Series and Functions
    • 10.16 Taylor Series
    • 10.17 Applications of Series
    • 10.18 Binomial Series
  • 11. Vectors
    • 11.1 Vectors - The Basics
    • 11.2 Vector Arithmetic
    • 11.3 Dot Product
    • 11.4 Cross Product
  • 12. 3-Dimensional Space
    • 12.1 The 3-D Coordinate System
    • 12.2 Equations of Lines
    • 12.3 Equations of Planes
    • 12.4 Quadric Surfaces
    • 12.5 Functions of Several Variables
    • 12.6 Vector Functions
    • 12.7 Calculus with Vector Functions
    • 12.8 Tangent, Normal and Binormal Vectors
    • 12.9 Arc Length with Vector Functions
    • 12.10 Curvature
    • 12.11 Velocity and Acceleration
    • 12.12 Cylindrical Coordinates
    • 12.13 Spherical Coordinates
• Calculus III
  • 12. 3-Dimensional Space
    • 12.1 The 3-D Coordinate System
    • 12.2 Equations of Lines
    • 12.3 Equations of Planes
    • 12.4 Quadric Surfaces
    • 12.5 Functions of Several Variables
    • 12.6 Vector Functions
    • 12.7 Calculus with Vector Functions
    • 12.8 Tangent, Normal and Binormal Vectors
    • 12.9 Arc Length with Vector Functions
    • 12.10 Curvature
    • 12.11 Velocity and Acceleration
    • 12.12 Cylindrical Coordinates
    • 12.13 Spherical Coordinates
  • 13. Partial Derivatives
    • 13.1 Limits
    • 13.2 Partial Derivatives
    • 13.3 Interpretations of Partial Derivatives
    • 13.4 Higher Order Partial Derivatives
    • 13.5 Differentials
    • 13.6 Chain Rule
    • 13.7 Directional Derivatives
  • 14. Applications of Partial Derivatives
    • 14.1 Tangent Planes and Linear Approximations
    • 14.2 Gradient Vector, Tangent Planes and Normal Lines
    • 14.3 Relative Minimums and Maximums
    • 14.4 Absolute Minimums and Maximums
    • 14.5 Lagrange Multipliers
  • 15. Multiple Integrals
    • 15.1 Double Integrals
    • 15.2 Iterated Integrals
    • 15.3 Double Integrals over General Regions
    • 15.4 Double Integrals in Polar Coordinates
    • 15.5 Triple Integrals
    • 15.6 Triple Integrals in Cylindrical Coordinates
    • 15.7 Triple Integrals in Spherical Coordinates
    • 15.8 Change of Variables
    • 15.9 Surface Area
    • 15.10 Area and Volume Revisited
  • 16. Line Integrals
    • 16.1 Vector Fields
    • 16.2 Line Integrals - Part I
    • 16.3 Line Integrals - Part II
    • 16.4 Line Integrals of Vector Fields
    • 16.5 Fundamental Theorem for Line Integrals
    • 16.6 Conservative Vector Fields
    • 16.7 Green's Theorem
  • 17.Surface Integrals
    • 17.1 Curl and Divergence
    • 17.2 Parametric Surfaces
    • 17.3 Surface Integrals
    • 17.4 Surface Integrals of Vector Fields
    • 17.5 Stokes' Theorem
    • 17.6 Divergence Theorem
• Differential Equations
  • 1. Basic Concepts
    • 1.1 Definitions
    • 1.2 Direction Fields
    • 1.3 Final Thoughts
  • 2. First Order DE's
    • 2.1 Linear Equations
    • 2.2 Separable Equations
    • 2.3 Exact Equations
    • 2.4 Bernoulli Differential Equations
    • 2.5 Substitutions
    • 2.6 Intervals of Validity
    • 2.7 Modeling with First Order DE's
    • 2.8 Equilibrium Solutions
    • 2.9 Euler's Method
  • 3. Second Order DE's
    • 3.1 Basic Concepts
    • 3.2 Real & Distinct Roots
    • 3.3 Complex Roots
    • 3.4 Repeated Roots
    • 3.5 Reduction of Order
    • 3.6 Fundamental Sets of Solutions
    • 3.7 More on the Wronskian
    • 3.8 Nonhomogeneous Differential Equations
    • 3.9 Undetermined Coefficients
    • 3.10 Variation of Parameters
    • 3.11 Mechanical Vibrations
  • 4. Laplace Transforms
    • 4.1 The Definition
    • 4.2 Laplace Transforms
    • 4.3 Inverse Laplace Transforms
    • 4.4 Step Functions
    • 4.5 Solving IVP's with Laplace Transforms
    • 4.6 Nonconstant Coefficient IVP's
    • 4.7 IVP's With Step Functions
    • 4.8 Dirac Delta Function
    • 4.9 Convolution Integrals
    • 4.10 Table Of Laplace Transforms
  • 5. Systems of DE's
    • 5.1 Review : Systems of Equations
    • 5.2 Review : Matrices & Vectors
    • 5.3 Review : Eigenvalues & Eigenvectors
    • 5.4 Systems of Differential Equations
    • 5.5 Solutions to Systems
    • 5.6 Phase Plane
    • 5.7 Real Eigenvalues
    • 5.8 Complex Eigenvalues
    • 5.9 Repeated Eigenvalues
    • 5.10 Nonhomogeneous Systems
    • 5.11 Laplace Transforms
    • 5.12 Modeling
  • 6. Series Solutions to DE's
    • 6.1 Review : Power Series
    • 6.2 Review : Taylor Series
    • 6.3 Series Solutions
    • 6.4 Euler Equations
  • 7. Higher Order Differential Equations
    • 7.1 Basic Concepts for nth Order Linear Equations
    • 7.2 Linear Homogeneous Differential Equations
    • 7.3 Undetermined Coefficients
    • 7.4 Variation of Parameters
    • 7.5 Laplace Transforms
    • 7.6 Systems of Differential Equations
    • 7.7 Series Solutions
  • 8. Boundary Value Problems & Fourier Series
    • 8.1 Boundary Value Problems
    • 8.2 Eigenvalues and Eigenfunctions
    • 8.3 Periodic Functions & Orthogonal Functions
    • 8.4 Fourier Sine Series
    • 8.5 Fourier Cosine Series
    • 8.6 Fourier Series
    • 8.7 Convergence of Fourier Series
  • 9. Partial Differential Equations
    • 9.1 The Heat Equation
    • 9.2 The Wave Equation
    • 9.3 Terminology
    • 9.4 Separation of Variables
    • 9.5 Solving the Heat Equation
    • 9.6 Heat Equation with Non-Zero Temperature Boundaries
    • 9.7 Laplace's Equation
    • 9.8 Vibrating String
    • 9.9 Summary of Separation of Variables
• Extras
• Algebra & Trig Review
  • 1. Algebra
    • 1.1 Exponents
    • 1.2 Absolute Value
    • 1.3 Radicals
    • 1.4 Rationalizing
    • 1.5 Functions
    • 1.6 Multiplying Polynomials
    • 1.7 Factoring
    • 1.8 Simplifying Rational Expressions
    • 1.9 Graphing and Common Graphs
    • 1.10 Solving Equations, Part I
    • 1.11 Solving Equations, Part II
    • 1.12 Solving Systems of Equations
    • 1.13 Solving Inequalities
    • 1.14 Absolute Value Equations and Inequalities
  • 2. Trigonometry
    • 2.1 Trig Function Evaluation
    • 2.2 Graphs of Trig Functions
    • 2.3 Trig Formulas
    • 2.4 Solving Trig Equations
    • 2.5 Inverse Trig Functions
  • 3. Exponentials & Logarithms
    • 3.1 Basic Exponential Functions
    • 3.2 Basic Logarithm Functions
    • 3.3 Logarithm Properties
    • 3.4 Simplifying Logarithms
    • 3.5 Solving Exponential Equations
    • 3.6 Solving Logarithm Equations
• Common Math Errors
  • 1. General Errors
  • 2. Algebra Errors
  • 3. Trig Errors
  • 4. Common Errors
  • 5. Calculus Errors
• Complex Number Primer
  • 1. The Definition
  • 2. Arithmetic
  • 3. Conjugate and Modulus
  • 4. Polar and Exponential Forms
  • 5. Powers and Roots
• How To Study Math
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Paul's Online Notes Home / Algebra / Solving Equations and Inequalities / Solutions and Solution Sets Prev. Section Notes Practice Problems Assignment Problems Next Section Show Mobile Notice Show All Notes Hide All Notes Mobile Notice You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 2.1 : Solutions and Solution Sets

We will start off this chapter with a fairly short section with some basic terminology that we use on a fairly regular basis in solving equations and inequalities.

First, a solution to an equation or inequality is any number that, when plugged into the equation/inequality, will satisfy the equation/inequality. So, just what do we mean by satisfy? Let’s work an example or two to illustrate this.

Example 1 Show that each of the following numbers are solutions to the given equation or inequality.

1. \(x = 3\) in \({x^2} - 9 = 0\)
2. \(y = 8\) in \(3\left( {y + 1} \right) = 4y - 5\)
3. \(z = 1\) in \(2\left( {z - 5} \right) \le 4z\)
4. \(z = - 5\) in \(2\left( {z - 5} \right) \le 4z\)

Show All Solutions Hide All Solutions a \(x = 3\) in \({x^2} - 9 = 0\) Show Solution

We first plug the proposed solution into the equation.

\[\begin{align*}{3^2} - 9 & \mathop = \limits^? 0\\ 9 - 9 & = 0\\ 0 & = 0 \,\,\,\,{\mbox{OK}}\end{align*}\]

So, what we are asking here is does the right side equal the left side after we plug in the proposed solution. That is the meaning of the “?” above the equal sign in the first line.

Since the right side and the left side are the same we say that \(x = 3\) satisfies the equation.

b \(y = 8\) in \(3\left( {y + 1} \right) = 4y - 5\) Show Solution

So, we want to see if \(y = 8\) satisfies the equation. First plug the value into the equation.

\[\begin{align*}3\left( {8 + 1} \right) &\mathop = \limits^? 4\left( 8 \right) - 5\\ 27 & = 27\,\,\,\,{\mbox{OK}}\end{align*}\]

So, \(y = 8\) satisfies the equation and so is a solution.

c \(z = 1\) in \(2\left( {z - 5} \right) \le 4z\) Show Solution

In this case we’ve got an inequality and in this case “satisfy” means something slightly different. In this case we will say that a number will satisfy the inequality if, after plugging it in, we get a true inequality as a result.

Let’s check \(z = 1\).

\[\begin{align*}2\left( {1 - 5} \right) & \mathop \le \limits^? 4\left( 1 \right)\\ - 8 & \le 4 \,\,\,\,{\mbox{OK}}\end{align*}\]

So, -8 is less than or equal to 4 (in fact it’s less than) and so we have a true inequality. Therefore \(z = 1\) will satisfy the inequality and hence is a solution

d \(z = - 5\) in \(2\left( {z - 5} \right) \le 4z\) Show Solution

This is the same inequality with a different value so let’s check that.

\[\begin{align*}2\left( { - 5 - 5} \right) & \mathop \le \limits^? 4\left( { - 5} \right)\\ - 20 & \le - 20\,\,\,\,{\mbox{OK}}\end{align*}\]

In this case -20 is less than or equal to -20 (in this case it’s equal) and so again we get a true inequality and so \(z = - 5\) satisfies the inequality and so will be a solution.

We should also do a quick example of numbers that aren’t solution so we can see how these will work as well.

Example 2 Show that the following numbers aren’t solutions to the given equation or inequality.

1. \(y = - 2\) in \(3\left( {y + 1} \right) = 4y - 5\)
2. \(z = - 12\) in \(2\left( {z - 5} \right) \le 4z\)

Show All Solutions Hide All Solutions a \(y = - 2\) in \(3\left( {y + 1} \right) = 4y - 5\) Show Solution

In this case we do essentially the same thing that we did in the previous example. Plug the number in and show that this time it doesn’t satisfy the equation. For equations that will mean that the right side of the equation will not equal the left side of the equation.

\[\begin{align*}3\left( { - 2 + 1} \right) & \mathop = \limits^? 4\left( { - 2} \right) - 5\\ - 3 & \ne - 13\,\,\,\,{\mbox{NOT OK}}\end{align*}\]

So, -3 is not the same as -13 and so the equation isn’t satisfied. Therefore \(y = - 2\) isn’t a solution to the equation.

b \(z = - 12\) in \(2\left( {z - 5} \right) \le 4z\) Show Solution

This time we’ve got an inequality. A number will not satisfy an inequality if we get an inequality that isn’t true after plugging the number in.

\[\begin{align*}2\left( { - 12 - 5} \right) & \mathop \le \limits^? 4\left( { - 12} \right)\\ - 34\require{cancel} & \bcancel{ \le } - 48\,\,\,\,{\mbox{NOT OK}}\end{align*}\]

In this case -34 is NOT less than or equal to -48 and so the inequality isn’t satisfied. Therefore \(z = - 12\) is not a solution to the inequality.

Now, there is no reason to think that a given equation or inequality will only have a single solution. In fact, as the first example showed the inequality \(2\left( {z - 5} \right) \le 4z\) has at least two solutions. Also, you might have noticed that \(x = 3\) is not the only solution to \({x^2} - 9 = 0\). In this case \(x = - 3\) is also a solution.

We call the complete set of all solutions the solution set for the equation or inequality. There is also some formal notation for solution sets although we won’t be using it all that often in this course. Regardless of that fact we should still acknowledge it.

For equations we denote the solution set by enclosing all the solutions is a set of braces, \(\left\{ {} \right\}\). For the two equations we looked at above here are the solution sets.

\[\begin{align*}3\left( {y + 1} \right) & = 4y - 5 & \hspace{0.25in} & {\mbox{Solution Set }} :\,\,\,\left\{ 8 \right\}\\ {x^2} - 9 & = 0 & \hspace{0.25in} & {\mbox{Solution Set }} :\,\,\,\left\{ { - 3,3} \right\}\end{align*}\]

For inequalities we have a similar notation. Depending on the complexity of the inequality the solution set may be a single number or it may be a range of numbers. If it is a single number then we use the same notation as we used for equations. If the solution set is a range of numbers, as the one we looked at above is, we will use something called set builder notation. Here is the solution set for the inequality we looked at above.

\[\left\{ {z|z \ge - 5} \right\}\]

This is read as : “The set of all \(z\) such that \(z\) is greater than or equal to -5”.

Most of the inequalities that we will be looking at will have simple enough solution sets that we often just shorthand this as,

\[z \ge - 5\]

There is one final topic that we need to address as far as solution sets go before leaving this section. Consider the following equation and inequality.

\[\begin{align*}{x^2} + 1 & = 0\\ {x^2} & < 0\end{align*}\]

If we restrict ourselves to only real solutions (which we won’t always do) then there is no solution to the equation. Squaring \(x\) makes \(x\) greater than equal to zero, then adding 1 onto that means that the left side is guaranteed to be at least 1. In other words, there is no real solution to this equation. For the same basic reason there is no solution to the inequality. Squaring any real \(x\) makes it positive or zero and so will never be negative.

We need a way to denote the fact that there are no solutions here. In solution set notation we say that the solution set is empty and denote it with the symbol : \(\emptyset \). This symbol is often called the empty set.

We now need to make a couple of final comments before leaving this section.

In the above discussion of empty sets we assumed that we were only looking for real solutions. While that is what we will be doing for inequalities, we won’t be restricting ourselves to real solutions with equations. Once we get around to solving quadratic equations (which \({x^2} + 1 = 0\) is) we will allow solutions to be complex numbers and in the case looked at above there are complex solutions to \({x^2} + 1 = 0\). If you don’t know how to find these at this point that is fine we will be covering that material in a couple of sections. At this point just accept that \({x^2} + 1 = 0\) does have complex solutions.

Finally, as noted above we won’t be using the solution set notation much in this course. It is a nice notation and does have some use on occasion especially for complicated solutions. However, for the vast majority of the equations and inequalities that we will be looking at will have simple enough solution sets that it’s just easier to write down the solutions and let it go at that. Therefore, that is what we will not be using the notation for our solution sets. However, you should be aware of the notation and know what it means.

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