equality

Examples of equality in the following topics:

• Equations and Inequalities

  • An equation states that two expressions are equal, while an inequality relates two different values.
  • An equation is a mathematical statement that asserts the equality of two expressions.
  • This is written by placing the expressions on either side of an equals sign (=), for example:
  • The notation $a \neq b$ means that a is not equal to $b$.
  • In either case, $a$ is not equal to $b$.

• Linear Inequalities

  • A linear inequality is an expression that is designated as less than, greater than, less than or equal to, or greater than or equal to.
  • When two linear expressions are not equal, but are designated as less than ($<$), greater than ($>$), less than or equal to ($\leq$) or greater than or equal to ($\geq$), it is called a linear inequality.  
  • A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.
  • A linear inequality looks like a linear equation, with the inequality sign replacing the equal sign.  
  • This reads "$x$ is greater than or equal to 6".

• Solving Problems with Rational Functions

  • The $x$-intercepts of rational functions are found by setting the polynomial in the numerator equal to $0$ and solving for $x$.
  • In the case of rational functions, the $x$-intercepts exist when the numerator is equal to $0$.
  • In order to solve rational functions for their $x$-intercepts, set the polynomial in the numerator equal to zero, and solve for $x$ by factoring where applicable.
  • Set the numerator of this rational function equal to zero and solve for $x$:
  • Here, the numerator is a constant, and therefore, cannot be set equal to $0$.

• Rational Equations

  • A rational equation sets two rational expressions equal to each other and involves unknown values that make the equation true.
  • Notice that the rational expressions on both sides of the equal sign have the same denominator.
  • Two triangles are said to be "similar" if they have equal corresponding angles. 
  • This is the same as the triangles having equal side-length ratios.
  • For example, $\angle A$ (i.e., angle $A$) corresponds to $\angle E$, and they are equal.

• Solving Equations: Addition and Multiplication Properties of Equality

  • The addition and multiplication properties of equalities are useful tools for solving equations.
  • In modern notation, this is indicated by placing the expressions on either side of an equal sign (=).
  • For example, $x+3=5$ asserts that $x+3$ is equal to 5.
  • Let $x$ equal the unknown value: the number of hours of labor.
  • In English, the cost of the labor ($34) multiplied by the number of hours of labor$(x)$, plus the cost of the parts ($339), is equal to the total bill for the repair ($458).

• Finding Polynomials with Given Zeros

  • To construct a polynomial from given zeros, set $x$ equal to each zero, move everything to one side, then multiply each resulting equation.
  • If we already count multiplicity in this number, than the degree equals the number of roots.
  • In the picture below, the blue graph represents the solution for $c$ equal to $1$.
  • The red graph represents the solution for $c$ equal to $-1/2$.
  • They are equal up to a constant.

• What is an Equation?

  • These are still written by placing each expression on either side of an equals sign ($=$).
  • For example, the equation $x + 3 = 5$, read "$x$ plus three equals five", asserts that the expression $x+3$ is equal to the value 5.
  • We use an equals sign to show that we know the value of a given variable.
  • In this case, we write $x=2$ (read as "$x$ equals two").
  • This equality is a true statement.

• Introduction to Inequalities

  • In the same way that equations use an equals sign, =, to show that two values are equal, inequalities use signs to show that two values are not equal and to describe their relationship.
  • Strict inequalities differ from the notation $a \neq b$, which means that a is not equal to $b$.
  • In the two types of strict inequalities, $a$ is not equal to $b$.
  • The notation $a \leq b$ means that $a$ is less than or equal to $b$ (or, equivalently, "at most" $b$).
  • The notation $a \geq b$ means that $a$ is greater than or equal to $b$ (or, equivalently, "at least" $b$).

• Factoring General Quadratics

  • When $a$ is equal to one, $\alpha_1$ and $\alpha_2$ both equal one, and $\beta_1$ and $\beta_2$ are factors of the constant $c$ such that:
  • When $a$ is not equal to one and not equal to zero, you can FOIL the above expression for the factored form of the quadratic to find that  $\alpha_1$ and $\alpha_2$ are factors of $a$ such that:
  • Next we need to find the factored set of values that add to equal the value of $b$.  
  • In this case, the correct values are 3 and -7, since they add to equal $-4$.  

• Graphs of Linear Inequalities

  • The simplest inequality to graph is a single inequality in two variables, usually of the form: $y\leq mx+b$, where the inequality can be of any type, less than, less than or equal to, greater than, greater than or equal to, or not equal to.
  • Since the equation is less than or equal to, start off by drawing the line $y=x+2$, using a solid line.  
  • Next, note that $y$ is less than or equal to $x+2$, which means that $y$ can take on the values along the line ("or equal to"), or any values below the line ("less than"), and so we shade in all the values under the line to get the following graph below.