Section 2.4 Applications of Linear Equations: Introduction to Problem Solving 149 Step 5: Solve the equation. 21x 62 3x 2 2x 12 3x 2 2x 2x 12 3x 2x 2 12 x 2 12 2 x 2 2 10 x The number is 10. Step 6: Write the final answer in words. 3. Consecutive Integer Problems The word consecutive means “following one after the other in order without gaps.” The numbers 6, 7, and 8 are examples of three consecutive integers. The numbers are examples of consecutive even integers. The numbers 23, 25, 4, 2, 0, and 2 and 27 are examples of consecutive odd integers. Notice that any two consecutive integers differ by 1. Therefore, if x represents an integer, then 1x 12 represents the next larger consecutive integer (Figure 2-4). Consecutive integers differ by 1 unit. x (x 1) 3 2 1 0 1 2 3 4 1 unit Figure 2-4 Instructor Note: Explain to students that we could let x be the larger of two consecutive integers. Then x 1 is the next smaller integer. However, using x and x 1 generally causes less confusion for students. Any two consecutive even integers differ by 2. Therefore, if x represents an even integer, then 1x 22 represents the next consecutive larger even integer (Figure 2-5). Consecutive even integers differ by 2 units. 3 2 1 0 1 2 3 4 Figure 2-5 x (x 2) 2 units Likewise, any two consecutive odd integers differ by 2. If x represents an odd integer, then 1x 22 is the next larger odd integer (Figure 2-6). Consecutive odd integers differ by 2 units. x (x 2) 3 2 1 0 1 2 3 4 2 units Figure 2-6 Avoiding Mistakes It is important to enclose “the sum of a number and six” within parentheses so that the entire quantity is multiplied by 2. Forgetting the parentheses would imply that only the x-term is multiplied by 2. Correct: 21x 62
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