Topic Overview: Part 1: Linear Functions Part 2: Linear Equations and Inequalities

Part 2: Linear Equations and Inequalities

Solving linear equations and inequalities is typically a large part of the Algebra 1 curriculum. The mathematical sentence 0.24x + 0.85 = 6.13 is an example of a linear equation; 0.5H - 450 100 is an example of a linear inequality.

Explanation

Linear equations and inequalities can help solve a wide range of problems, including predicting the cost of a phone call that takes a given number of minutes and predicting the number of hot dog sales necessary to make a profit of \$100, or a profit of at least \$250.

Other examples:
• To find the Celsius equivalent of 77 degrees Fahrenheit, solve the linear equation 77 = 1.8C + 32.

• If a utility company plan uses the formula C = 0.02P + 25 to calculate monthly charges, a family can determine how many units of power they can consume to pay no more than \$50 per month by solving 0.02P + 25 50.

• If a car salesman makes a monthly base salary of \$1,500 and receives an additional \$400 dollars for each car sold, the equation 400C + 1500 4,700 gives the number of cars that he must sell to earn a monthly income of at least \$4,700.

In many classrooms, students learn only the algebraic manipulations necessary to solve problems. It is important for them to know that they can also solve linear equations and inequalities using tables and graphs. Below, note the three different ways of solving the equation 0.24x + 0.85 = 6.13. In this problem, x represents the length (in minutes) of a phone call that costs a total of \$6.13. To use the table and graph method, let y = 0.24x + 0.85 where y represents the total cost of a call lasting x minutes.

 Algebraically Table Graph Minutes Cost 18 5.17 19 5.41 20 5.65 21 5.89 22 6.13 23 6.37 25 6.61

Conclusion: A 22-minute phone call costs \$6.13.

Solving Algebraically
Students solve the equation by choosing equivalence transformations to apply to both sides of the equation. First, they subtract 0.85 from both sides of the equation and simplify the results. Then, they divide both sides by 0.24 and simplify the results. It is important to note that this process only works for linear equations. When students study other functions, they must learn the algebraic techniques that allow them to produce the solution to that particular function.

Solving With Tables
Students enter the equation y = 0.24x + 0.85 into a graphing calculator. Looking at the table of values, they can identify the x-value that produces the desired y-value of 6.13. That occurs when the x-value is 22. Essentially, the table is a quick and efficient guess-and-check method of solution that works for any type of function. To refine guesses, students can adjust the increments in the table.

Solving With Graphs
Students enter the equation y = 0.24x + 0.85 into the calculator. Looking at the graph of the line, they can identify 22 as the x-value that produces the desired y-value of 6.13. One way to find the desired value is to use the calculator's trace function to follow the line to the desired point. Like the table, the graph is a quick and efficient guess-and-check method of solution that works for any type of function. To refine guesses, students can also graph y = 6.13 and find the x-value where the two lines intersect.
While students should understand and make connections between all three solution methods, the tabular and graphic methods are particularly helpful to students who find symbolic manipulation difficult. (This will be explored fully in the "Rule of Four" section of Workshop 5.)

Mathematical Definition

A linear equation is an equation that can be written in the form Ax + By = C and whose graph is a straight line. An inequality is an open sentence that contains the symbol , , >, or <.

The following definitions come from a mathematics dictionary:

Linear equation or expression: An algebraic equation or expression which is of the first degree in its variable (or variables); i.e., its highest degree term in the variable (or variables) is of the first degree. The equations x + 2 = 0 and x + y + 3 = 0 are linear.

Inequality: A statement that one quantity is less than (or greater than) another. If a is less than b, their relation is denoted symbolically by a < b; the relation a greater than b is written a > b. Inequalities have many important properties ... An inequality which is not true for all values of the variables involved is a conditional inequality; e.g., (x + 2) > 3 is a conditional inequality, because it is true only for x [values] greater than 1.
(Source: James, Robert C. and Glenn James. Mathematics Dictionary [4th edition]. New York: Van Nostrand Reinhold, 1976.)

Role in the Curriculum

Solving linear equations and inequalities is a major focus in algebra. Students should be proficient in solving equations and inequalities algebraically, as well as with tables and graphs. According to the National Council of Teachers of Mathematics (NCTM):

Algebra ... should provide students with insights into mathematical abstraction and structure. Students should develop an understanding of the algebraic properties that govern that manipulation of symbols in expressions, equations, and inequalities. They should become fluent in performing such manipulations by appropriate means - mentally, by hand, or by machine - to solve equations and inequalities, to generate equivalent forms of expressions or functions, or to prove general results.
(NCTM, Principles and Standards for School Mathematics, 2000, p. 297)
NCTM further states that students need a solid understanding of the order of operations and the distributive, associative, and commutative properties in order to become fluent with symbol manipulation (PSSM, 2000, p. 227). See the Workshop 2 video for examples of how two algebra teachers promote this in their classrooms.

Read Fran Curcio's thoughts about the important aspects of solving equations and inequalities that Janel Green applies in her lesson in the video for Workshop 2, Part II:

 Read transcript from teacher educator Fran Curcio Mathematics is such a wonderful subject that allows students to explore expressions and relationships in a variety of ways. Read More

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