# Chapter 1 Section 5: Equations and Inequalities involving Absolute Value

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1 Introduction The concept of absolute value is very strongly connected to the concept of distance. The absolute value of a number is that number s distance from 0 on the number line. Since distance is always positive (or zero), the absolute value is always positive (or zero). Seeing absolute values as distances will make working with them much easier and more intuitive. Note: Watch the animation in the course online to review the concept of absolute value. Example A Solve the equation x 3. Think of the equation as asking What points x on the number line are at a distance of three from zero? Look at the number line: You can see that both 3 and 3 are the required distance from zero. These are the two solutions 3, 3. to the equation above, so the solution set for the equation is { } Solving A B (with B > 0 ) In general, whenever you have an equation of the form A B, with B > 0, then the solutions can be found by solving the following equations: A B and A B. (Note that here A and B represent any algebraic expressions.) The above is true only if B > 0. If B 0, then the only solutions occur when A 0. If B < 0, then there can be no solution because in that case, the left side of the equal sign is positive while the right side is negative (see Example B). No negative number equals any positive number! Example B Solve: x 2. x 2 + Recall that the absolute value of x is a distance, and so it is positive (or zero). And, of course, a positive number can never equal a negative number. So the solution set for this equation is the empty set,. Page 1 of 15

2 Example C Solve: 6x 9 3. Since 3 > 0, the two solutions to this equation can be found by solving the equations: 6x 9 3 and 6x 9 3. Solve the first equation by adding 9 to both sides and then dividing by 6: 6x x 12 6 x x 2 The second equation is also solved by adding 9 to both sides and then dividing by 6: 6x x 6 6 x x 1 The solution set is the set containing the two solutions found: { 1, 2 }. Extended Example 1a Solve: 2x Find the solution set of this equation. Hint: Solve the equation 2x x x 2 2 x x 1 Hint: Solve the equation 2x x x 12 2 x x 6 Hint: The solution set of the equation is the set containing the two solutions you ve just found. 6, 1. The solution set is { } Page 2 of 15

3 Extended Example 1b Solve: 5x 1 9. Find the solution set of this equation. Hint: Can the non-negative quantity on the left side of the equal sign possibly equal the negative quantity on the right side? Since no positive number equals 9, this equation has no solution. The solution set is the empty set,. Extended Example 1c Solve: 7 8. Find the solution set of this equation. Hint: Solve the equation x x 3 Hint: Solve the equation x x 5 Hint: The solution set of the equation is the set containing the two solutions you ve just found. 1 The solution set is,5 3. Page 3 of 15

4 Extended Example 1d Solve: 8x Find the solution set of this equation. Hint: Solve the equation 8x x x 18 8 x x 4 Hint: Solve the equation 8x x x 4 8 x x 2 Hint: The solution set of the equation is the set containing the two solutions you ve just found. 1 9 The solution set is, 2 4. Extended Example 1e Solve: y 1. Find the solution set of this equation. Hint: Can the non-negative quantity on the left side of the equal sign possibly equal the negative quantity on the right side? Since no number greater than or equal to zero can possibly equal 1, this equation can have no solution. The solution set is the empty set,. Page 4 of 15

5 Solving A B The solutions to an equation of the form A B can be found by solving the equations: A B and A B. If B is the opposite of A, then the absolute value of B will equal the absolute value of A (since the absolute value will get rid of the negative). Once again, it helps to think of this absolute value equation in terms of distance. The equation says that the distance from A to 0 is the same as the distance from B to 0. So, either A and B are the same point, or they are opposites on the number line (see Example D). Example D Solve: 5x 9 3 7x. Solutions of this equation can be found by solving equations 5x 9 3 7x and 5x x. ( ) To solve the first equation, first add 7x to both sides and add 9 to both sides: 5x 9 3 7x + 7x x 12x 12 Then, divide both sides by 12: 12 x x 1 To solve the second one, first distribute the negative sign on the right side: ( x) 5x x x. Then subtract 5x and add 3 to both sides: 5x x 5x x 6 2x Divide both sides by 2: 6 2 x x So, the solution set of our absolute value equation is { 3, 1}. Extended Example 2a Solve: 8x 1 8. Find the solution set of this equation. Hint: Solve the equation 8x 1 8, by first adding and 1 to both sides. Page 5 of 15

6 Extended Example 2a, 8x x 9 11 x x 11 Hint: Solve the equation 8x 1 ( 8 ), by first distributing the negative to eliminate parentheses, and then subtracting while adding 1to both sides. 8x 1 8 ( ) 8x x 7 5 x x 5 Hint: The solution set of the equation is the set containing the two solutions you ve just found. 7 9 The solution set is, Extended Example 2b Solve: 5y + 8 9y 2. Find the solution set of this equation. Hint: Solve the equation 5y + 8 9y 2, by first subtracting 5y and adding 2 to both sides. 5y + 8 9y 2 5y + 2 5y y y y Page 6 of 15

7 Extended Example 2b, +, by first distributing the negative to eliminate parentheses, and then adding 9y while subtracting 8 from both sides. 5y + 8 ( 9y 2) 5y + 8 9y y 8 + 9y 8 14y 6 14 y y 7 Hint: Solve the equation 5y 8 ( 9y 2) Hint: The solution set of the equation is the set containing the two solutions you ve just found. 3 5 The solution set is, 7 2. Extended Example 2c Solve: y 12 4y 1. Find the solution set of this equation. Hint: Solve the equation y 12 4y 1, by first subtracting y while adding 1to both sides. y 12 4y 1 y + 1 y y 11 3 y y 3, by first distributing the negative to eliminate parentheses, and then adding 4y and 12to both sides. y 12 ( 4y 1) y 12 4y y y y 13 5 y Hint: Solve the equation y 12 ( 4y 1) y 5 Page 7 of 15

8 Extended Example 2c, Hint: The solution set of the equation is the set containing the two solutions you ve just found The solution set is, 3 5. Solving A < B (with B > 0 ) To solve an inequality of the form A < B, with B > 0, just solve the equivalent three-way inequality: B < A< B. The original inequality says that the distance from A to 0 is less than B. A graph will help you visualize this: So the solution set is the interval ( B B) negative number (or 0 ) is greater than any positive number.,. Note that if B 0, then there can be no solution. No Example E Solve: x < 2. This inequality has no solution, since the left side is positive (or zero), while the right side is negative: x < 2 never + < The solution set for this equation is thus the empty set,. Solving A B (with B 0 ) To solve an inequality of the form A B, with B 0, just solve the equivalent three-way inequality: B A B. In this case, B can be zero, as long as A is also zero. Page 8 of 15

9 Extended Example 3a Solve: 8x Express your solution in set-builder notation and in interval notation. Hint: Write this as a 3-way inequality. 23 8x Hint: Subtract 5 from all three sides. 23 8x x 18 Hint: Divide all three sides by 8. Step 3: 28 8x x x 2 4 Hint: Write the solution in set-builder and in interval notation x x, Extended Example 3b Solve: 6x 7 < 8. Find the solution set of this equation. Hint: Can the non-negative quantity on the left side of the equal sign possibly be less than the negative quantity on the right side? Since no number greater than or equal to zero can possibly be less than 8, this equation can have no solution. The solution set is the empty set,. Extended Example 3c Solve: 7x 6 < 15. Express your solution in set-builder notation and in interval notation. Hint: Write this as a 3-way inequality. 15 < 7x 6 < 15 Hint: Add 6 to all three sides. Page 9 of 15

10 Extended Example 3c, 15 < 7x 6 < < 7x < 21 Hint: Divide all three sides by 7. Step 3: 9 < 7x < x 21 < < x 7 3 < < < x < 3 7 Hint: Write the solution in set-builder and in interval notation. 9 9 x < x < 3, Extended Example 3d Solve: 10 11x < 21. Express your solution in set-builder notation and in interval notation. Hint: Write this as a 3-way inequality. 21 < 10 11x < 21 Hint: Subtract 10 from all three sides. 21 < 10 11x < < 11x < 11 Hint: Divide all three sides by 11. Remember to reverse the inequalities, since we re dividing by a negative. Step 3: 31 11x 11 > > x > > > x > 1 11 Page 10 of 15

11 Extended Example 3d, Hint: Write the solution in set-builder and in interval notation x 1< x < 1, Extended Example 3e Solve: 5x Find the solution set of this equation. Hint: Write this as a 3-way inequality. 0 5x x Hint: The only way this can occur is when 5x + 3 0, because no number can be both greater than zero and less than zero. Solve for x. 5x x 3 5 x x 5 Hint: Write down the solution set of the inequality, which has but one element. 3 5 Solving A > B (with B > 0 ) To solve an inequality of the form A inequalities: A > B or A < B. > B, with B > 0, take the union of the solutions to the two The original inequality is true whenever the distance from A to 0 is greater than B. A graph will help you visualize this: For example, the inequality x > 5 is true whenever the distance from x to 0 is larger than 5. This can happen if x is larger than 5, but can also happen if x is less than 5. Page 11 of 15

12 To see why, imagine for example that x 6. Then, indeed, 6 6 > 5. In general, the inequality A > B has solutions if A is large enough or far enough from 0 in either the positive or negative direction. Note that if B < 0, then every real value of A is a solution. Every positive number (or zero) is larger than any negative number. Example F Solve x > 2. All real numbers are solutions to this inequality: The solution set is thus the set of all real numbers, expressed in interval notation as (, ), and in set-builder notation as { x x } + x > 2 always < <. > Extended Example 4a Solve: Graph your solution, and express it in set-builder notation and in interval notation. Hint: Note the two inequalities without absolute values that will yield the desired solution set or Hint: Solve the first inequality. First, add 10 to both sides. Then, divide both sides by x x 3 Hint: First, graph this solution and then write it in set-builder notation and in interval notation. Step 3: x x, 3 3 Hint: Solve the second inequality. Add 10 to both sides and then divide both sides by 3. Page 12 of 15

13 Extended Example 4a, Step 4: Step 5: 6 3 x x 2 Hint: Graph and then write this solution in set-builder notation and in interval notation. { x x 2 } (, 2] Step 6: Hint: Graph the union of the two solution sets. Hint: Express the solution you graphed in interval and set-builder notation. 14 (, 2 ], { x x 2} x x x x 2 or x 3 3 Extended Example 4b Solve: 2x + 13 > 4. Find the solution set of this equation. Hint: All non-negative quantities are greater than any negative number. No matter what x is, the absolute value is greater than or equal to 0 and thus greater than 4.The,, and in set solution set is thus the set of all real numbers, expressed in interval notation as ( ) builder notation as { x x } < <. Extended Example 4c Solve: 7 2x 1. Graph your solution, and express it in set-builder notation and in interval notation. Hint: Note the two inequalities without absolute values that will yield the desired solution set. 7 2x 1 or 7 2x 1 Hint: Solve the first inequality. First, subtract 7 from both sides. Then, divide both sides by 2. Page 13 of 15

14 Extended Example 4c, 7 2x x 6 2x x 2 3 x 3 Step 3: Hint: First, graph this solution and then write it in set-builder notation and in interval notation. { x x 3 } (,3] Hint: Solve the second inequality, 7 2x 1. Subtract 7 from both sides, and then divide both sides by 2. Step 4: 7 2x 1 7 2x 2x 2 2 x 2 x Step 5: Hint: Graph and then write this solution in set-builder notation and in interval notation. { x x 4} [ 4, ) Step 6: Hint: Graph the union of the two solution sets. Hint: Express the solution you graphed in interval and set-builder notation.,3 4, ( ] [ ) { x x 3} { x x 4} { x x 3 or x 4} Page 14 of 15

15 Extended Example 4d Solve: 4x + 7 > 11. Graph your solution, and express it in set-builder notation and in interval notation. Hint: Note the two inequalities without absolute values that will yield the desired solution set. 4x + 7 > 11 or 4x + 7 < 11 Hint: Solve the first inequality. First, subtract 7 from both sides and then divide both sides by 4. 4x + 7 > x > 4 4 x 4 > 4 4 x > 1 Step 3: Hint: First, graph this solution and then write it in set-builder notation and in interval notation. { x x > 1} ( 1, ) Hint: Solve the second inequality, 4x + 7 < 11. Subtract 7 from both sides and then divide both sides by 4. Step 4: 4x + 7 < x < 18 4 x 2 9 < x < Step 5: 9 2 Hint: Graph and then write this solution in set-builder notation and in interval notation. { x x < 9 } (, 9 ) 2 2 Hint: Graph the union of the two solution sets. Step 6: Hint: Express the solution you graphed in interval and set-builder notation. (, 9 ) ( 1, ) 2 { x x < 9 } { x x > 1} { x x < 9 or x > } End of Lesson Page 15 of 15

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