Absolute Value Equations and Inequalities

Transcription

1 Absolute Value Equations and Inequalities In this unit, you will solve absolute value equations in one variable and graph their solutions on the number line. You will also solve absolute value inequalities in one variable and graph their solutions on the number line. Represent an Absolute Value Equation on a Number Line Key Terms The important word or term you will learn is: absolute value The absolute value of a number is the distance between the number and 0 on the number line. A distance is always greater than or equal to zero, so the absolute value of a number is always greater than or equal to zero. An absolute value equation of the form, where is any number, can have two possible solutions, one solution, or no solution. You can use a number line to represent the solution to an absolute value equation. The solution is represented by the point or points that satisfy the equation. You can also use set notation to represent the solution to an absolute value equation. To use set notation, use curly braces { } to write the set of numbers that are part of the solution. Skills In this unit you will learn to: Solve absolute value equations Solve absolute value inequalities Represent the solution to each equation on a number line and write the solution using set notation. c. To find the solution to the absolute value equation, find all points on the number line that are 5 units from zero. There are two points that are 5 units from zero, 5 and. So, the solution to the absolute value equation is or. Solution: {5, } To find the solution to the absolute value equation, find all points on the number line that are 0 units from zero. Zero is the only point that is 0 units from 0. So, the solution to the absolute value equation is. Solution: {0} c. To find the solution to the absolute value equation, find all points on the number line that are units from zero. Distance cannot be negative, so the equation has no solution. Solve an Absolute Value Equation Carnegie Learning Page 1

2 An absolute value equation may be written in the form, where is greater than 0. When solving an absolute value equation, consider the fact that the expression inside the absolute value symbol can be positive or negative. For instance, in the expression, can be 3 or. So, to solve the absolute value equation where is greater than 0, set equal to and set equal to. Then solve each equation. Solve the equation and represent the solution on a number line. In this problem, the absolute value of the expression is equal to 9, so the expression inside the absolute value symbols is equal to 9 or. Write and solve each equation for. = 9 or = = 3 or = = 1 or = So, the solutions to the absolute value equation are or. The graph of the solution set is shown below. To verify that the solutions are correct, substitute each solution into the original absolute value equation. The left and right sides of each equation should be equal. 9 = 9 or 9 = 9 The left and right sides are equal for each equation, so or are the solutions to the absolute value equation. Represent an Absolute Value Inequality on a Number Line Represent the solution for each absolute value inequality on a number line. > 4 2 To represent the solution on the number line, graph all points that are greater than 4 units from 0. All numbers greater than 4 are more than 4 units from 0, and all numbers less than are more than 4 units from 0. So, one part of the solution is the ray that represents the graph of > 4. The other part of the solution is the ray that represents the graph of <. So, the solution is > 4 or <. To represent the solution on the number line, graph all points that are less Carnegie Learning Page 2

3 than or equal to 2 units from 0. All numbers between and 2, including and 2, are less than or equal to 2 units from 0. So, the solution is the segment that represents the overlapping regions of the graphs of 2 and. So, the solution is 2. Represent an Absolute Value Inequality with a Non-Standard Solution on a Number Line Represent the solution for each absolute value inequality on a number line. < The inequality states that the absolute value of is greater than or equal to a negative number. Because the absolute value of a number is always greater than or equal to zero, then the absolute value of a number is always greater than a negative number. So, the inequality > is true for any value of. To represent the solution on the number line, graph the line representing all points on the number line. So, the solution is = all real numbers. The inequality states that the absolute value of is less than 0. The absolute value of a number is always greater than or equal to 0. So, the inequality < has no solution. Solve an Absolute Value Inequality Involving Less Than To solve an absolute value inequality that contains an expression within the absolute value, you must first rewrite the inequality as one of the following equivalent compound inequalities. Absolute Value Inequality Equivalent Compound Inequality > < or > or < < < Then, split the compound inequality into two simple inequalities. Solve each simple inequality by isolating the variable. Carnegie Learning Page 3

4 Solve the inequality and represent the solution on a number line. To solve, rewrite the inequality as an equivalent compound inequality. In this problem, the equivalent compound inequality is < < 5. Split the compound inequality into two simple inequalities, and solve each simple inequality. > and > 0 and < 10 < 0 and > So, the solutions to the absolute value inequality are < 0 and >, or < < 0. The graph of the solution set is shown below. So, all numbers greater than and less than 0 satisfy the absolute value inequality. For instance, is greater than and less than 0. Substitute into the original absolute value inequality. 3 The resulting statement is true. So, < < 0 is the solution to the absolute value inequality. Solve an Absolute Value Inequality Involving Greater Than Solve the inequality and represent the solution on a number line. To solve, rewrite the inequality as an equivalent compound inequality. In this problem, the equivalent compound inequality is. Solve each simple inequality. or or or 12 or 3 So, the solutions to the absolute value inequality are or 3. The graph of the solution set is shown below. So, all numbers less than or equal to or greater than or equal to 3 satisfy the absolute value inequality. For instance, 5 is greater than or equal to 3. Substitute 5 into the original absolute value inequality. Carnegie Learning Page 4

5 28 The resulting statement is true. So, or 3 is the solution to the absolute value inequality. Carnegie Learning Page 5