3.3 Expressions, Equations, and Inequalities
A numerical expression is a meaningful string of numbers, operation symbols, and/or grouping symbols. In early elementary grades, examples of numerical expressions include \[2+6+4, \quad 13-3+2, \quad \mbox{ or } \quad (4+6)+3.\] In upper elementary grades (3-6), students are expected to incorporate rational numbers and the operations of multiplication, division, and exponents in order to be comfortable working with such expressions as \[3 \times (10+4), \quad \frac{2}{3} \times (12+8), \quad \mbox{ or } \quad 2^3 + \frac{3+2}{7}.\]
In the transition from elementary school to secondary schools, students learn to add variables, symbols or letters that stand for any number within a specified range of numbers, to these expressions and thus work with algebraic expressions such as \[100-10\cdot (P-15)^3, \quad 3xy + 5(x-4)^2-7y^2, \quad 6\cdot (4x+3y), \quad \mbox{ or }\quad \frac{3x+2}{2x-4}.\]
Related Content Standards
(6.EE.2) Write, read, and evaluate expressions in which letters stand for numbers.
- Write expressions that record operations with numbers and with letters standing for numbers.
- Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.
- Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
In more advanced courses in secondary school mathematics, algebraic expressions can also include functions giving rise to expressions such as \[f(x-h)+k, \quad \sin(3x)-2, \quad \mbox{ or } \quad \ln(x+3).\]
The equivalence of numerical expressions is a key concept in the early elementary grades where students learn to compose and decompose numbers in different ways as they add and subtract numbers using different algorithms. In sixth grade, students are asked to determine the equivalence of two mathematical expressions. This definition that two algebraic expressions are equivalent if they generate the same number regardless of which number is substituted for the variable is one of the first places where students are pushed to move to the abstract realm of equivalence relations. We can verify that this definition of equivalent expressions is an equivalence relation using the fact that equality on the real numbers is an equivalence relation.
Related Content Standards
- (6.EE.3) Apply the properties of operations to generate equivalent expressions. %{}
- (6.EE.4) Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).
- (6.EE.5) Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
While substituting in values to determine if two expressions are equivalent is the definition of the equivalence, it is almost always impossible to substitute in all possible values for a variable, as there are infinitely many possibilities in most situations. Instead, we determine the equivalency of two expressions by applying various properties of the operations. For example, the distributive property shows us that \(3(x+y)\) is equivalent to \(3x+3y\). However, students have to be careful to consider the set of possible numbers for an expression when determining the equivalence of \(\sqrt{x^2}\) and \(x\), or \(\frac{3x^2}{x}\) and \(3x\).
An equation is a statement that a number or expression is equivalent to a different number or expression. An equation that is true for all values of the variable is called an identity. The associative, commutative, and distributive properties of the number systems \[\begin{align*} a+(b+c) &= (a+b)+c \\ a+b &= b+a \\ a \times (b+c) &= (a\times b)+(a\times c) \end{align*}\] are all common identities. Some other common identities in this regard are the trigonometric identity \[\begin{align*} \cos^2(\theta) + \sin^2(\theta) &= 1 \\ \sin(\alpha \pm \beta) &= \sin(\alpha)\cos(\beta)\pm \cos(\alpha)\sin(\beta) \\ \cos(2\theta) &= \cos^2(\theta)-\sin^2(\theta) \\ \sin(2 \theta) &= 2 \sin(\theta)\cos(\theta) \end{align*}\] that are used to simplify trigonometric expressions.
An equation that describes how multiple quantities vary together is usually called a formula. These can include the area formula for rectangles defined by \(\mbox{Area}=\mbox{width} \times \mbox{height}\) or Boyle’s law where pressure (\(P\)) is inversely proportional to volume (\(V\)), i.e. \(PV =k\) for some constant \(k\).
While the identity equations hold true irrespective of the values of the variables, it is often the case that equations hold for only some of the possible values for the variables, or possibly even none. We can then solve such equations by determining which values, called solutions to the equation, for the variables cause the equation to be a true statement.
Similar to equations, inequalities are statements that a number or expression is related to a different number or expression in a certain order relation. Some of these inequalities are identities or formulas, such as \(x\leq |x|, x\in\mathbb{R}\), while some have a solution set, such as \(x^2+y^2<1\).
Definition 3.7 Two equations (inequalities) are equivalent if they are true statements for the same values of the variables.
In the process of finding solutions to equations or inequalities, it is almost always helpful to rewrite an equation or inequality as an equivalent equation or inequality. For instance in an effort to find the solutions to \[\frac{x^2-10x+21}{3x-12} = \frac{x-5}{x-4}\] we want to find an expression that is equivalent to the left side of the equation. Such an expression could be \[\frac{(x-3)(x-7)}{3(x-4)},\] with the equivalency of the expressions verified using the distributive property. Since these two expressions are equivalent, they have the same values for every value of the variable. This means that the original equation is a true statement if, and only if, \[\frac{(x-3)(x-7)}{3(x-4)}=\frac{x-5}{x-4}\] is a true statement. We also know that if two numbers are the same, then \(3\) times each of the numbers are also the same. So the original statement is true if, and only if, the equivalent equation \[\frac{(x-3)(x-7)}{x-4}=\frac{3(x-5)}{x-4}\] is true. If \(x=4\), then the statement would have no meaning, as division by zero has no meaning. This means that we can eliminate \(4\) from the set of possible solutions. With \(x\neq 4\) we can multiply both sides of the equation by \((x-4)\) to generate the equivalent equation, \[(x-3)(x-7)=3(x-5) \quad \mbox{and} \quad x\neq 4.\] We can use the distributive property again to create an equivalent equation of \[x^2-10x+21=3x-15 \quad \mbox{and} \quad x\neq 4.\] Since we can also add the same expression, \(-3x+15\), to both sides of the equation and create an equivalent equation, we see that the original equation is equivalent to \[x^2-13x+36=0 \quad \mbox{and} \quad x\neq 4.\] Using the distributive property, we see that this equation is equivalent to \[(x-4)(x-9)=0 \quad \mbox{and} \quad x\neq 4.\] So the original equation is a true statement if, and only if, \((x-4)(x-9)=0 \quad \mbox{and} \quad x\neq 4\) is a true statement. This only occurs when the variable, \(x\), is equal to \(9\).
Through this process we can see how equivalence of equations is essential in the process of finding solutions to equations. This means that as students make the transition from elementary to secondary, it is important for teachers to be understanding of the challenge of thinking about equivalent expressions and to be explicit about this relationship.
3.3.1 Exercises
- How would you use the information from this section to respond to the following student questions? Give an answer in the context of your expected student population.
- What is the difference between an expression and equation?
- Why can we not solve an expression?
- What does it mean to “solve an equation”?
- What makes two expressions equivalent to each other?
- What is the difference between how the equality sign is used when solving an equation versus when working with expressions?
- Prove that the definition of equivalent equations forms an equivalency relation.