NAME DATE PERIOD. Study Guide and Intervention. Solving Radical Equations and Inequalities

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1 6-7 Study Guide and Intervention Solve Radical Equations The following steps are used in solving equations that have variables in the radicand. Some algebraic procedures may be needed before you use these steps. Step 1 Step 2 Step Step 4 Isolate the radical on one side of the equation. To eliminate the radical, raise each side of the equation to a power equal to the index of the radical. Solve the resulting equation. Check your solution in the original equation to make sure that you have not obtained any extraneous roots. Example 1 Solve 2 4x = 8. Example 2 2 4x = 8 Original equation 2 4x + 8 = 12 Add 4 to each side. 4x + 8 = 6 Isolate the radical. 4x + 8 = 6 Square each side. 4x = 28 x = 7 Divide each side by 4. Check 2 4(7) (6) = 8 The solution x = 7 checks. Exercises Solve each equation. Subtract 8 from each side. Solve x + 1 = 5x - 1. x + 1 = 5x - 1 Original equation x + 1 = 5x - 2 5x + 1 Square each side. 2 5x = 2x Simplify. 5x = x Isolate the radical. 5x = x 2 Square each side. x 2-5x = 0 x(x - 5) = 0 Factor. x = 0 or x = 5 Check x = x = x - 4 = x - 1 = x - 4 = x = 5 Subtract 5x from each side. (0) + 1 = 1, but 5(0) - 1 = -1, so 0 is not a solution. (5) + 1 = 4, and 5(5) - 1 = 4, so the solution is x = x + 1 = x = x = 7x - 9 Lesson x = x - 11 = x (9x - 11) 1 2 = x + 1 Chapter 6 45 Glencoe Algebra 2

2 6-7 Study Guide and Intervention (continued) Solve Radical Inequalities A radical inequality is an inequality that has a variable in a radicand. Use the following steps to solve radical inequalities. Step 1 If the index of the root is even, identify the values of the variable for which the radicand is nonnegative. Step 2 Solve the inequality algebraically. Step Test values to check your solution. Example Solve 5-20x Since the radicand of a square root must be greater than or equal to zero, first solve 20x x x -4 x Now solve 5-20x x Original inequality 20x x x 60 Isolate the radical. Eliminate the radical by squaring each side. Subtract 4 from each side. x Divide each side by 20. It appears that - 1 x is the solution. Test some values. 5 x = -1 x = 0 x = 4 20(-1) + 4 is not a real number, so the inequality is not satisfied. Therefore the solution x checks. Exercises Solve each inequality. 1. c x x < (0) + 4 =, so the inequality is satisfied. 2. 2x < x > x (4) , so the inequality is not satisfied.. 10x > x d d 5 Chapter 6 46 Glencoe Algebra 2

3 6-7 Skills Practice Solve each equation. 1. x = 5 2. x + = 7. 5 j = 1 4. v = y 1 2 = w = 4 7. b - 5 = 4 8. n + 1 = 5 Lesson r - 6 = p + 7 = k = (2d + ) 1 = 2 1. (t - ) 1 = (1-7u) 1 = z - 2 = z - 4 Solve each inequality x x + < r - > g + 1 = 2g c a x + 1 > 2. y r Chapter 6 47 Glencoe Algebra 2

4 6-7 Practice Solve each equation. 1. x = x =. 2p + = h - 2 = 0 5. c = h 1 2 = d + 2 = w - 7 = q - 4 = y = m = m = n = t - 8 = t = 16. (7v - 2) = (g + 1) = (6u - 5) = d - 5 = d x - 4 = 2x + 10 Solve each inequality. 2. a q c r - 6 = r 22. 2x + 5 = 2x z a - < x - 1 < STATISTICS Statisticians use the formula σ = v to calculate a standard deviation σ, where v is the variance of a data set. Find the variance when the standard deviation is GRAVITATION Helena drops a ball from 25 feet above a lake. The formula t = h describes the time t in seconds that the ball is h feet above the water. How many feet above the water will the ball be after 1 second? Chapter 6 48 Glencoe Algebra 2

5 6-7 Word Problem Practice Rational Equations and Inequalities 1. SIGNS A sign painter must spend $8 n to make n signs. How many signs can the painter make for $1200? 4. TETHERS A tether is being attached to a 25-foot pole in such a way that x + y = 50. By the Pythagorean Theorem, the distance y = x What must x be? 2. LATERAL AREA The lateral area of a cone with base radius r and height h is given by the formula L = πr r 2 + h 2. A cone has a lateral area of 65π square units and a base radius of 5 units. y 25 ft Lesson 6-7 h r x What is the height of the cone?. ORIGAMI Georgia wants to fold a square piece of paper into an equilateral triangle. She wants to locate the distance x up the side of the square where she can make the fold indicated by the dashed line in the figure so that a = b. From geometry class, she knows that a = 1 + x 2 and b = 2 (1 - x). So the equation she must solve is 1 + x 2 = 2 (1 - x). What is x? 45 b 5. RANGE NASA s Near-Earth Asteroid Tracking System tracks more than 00 asteroids. An asteroid is passing near Earth. If Earth is located at the origin of a coordinate plane, the path that the asteroid will trace out is given by y = 17 x, x > 0. One unit corresponds to one million miles. Carl learns that he will be able to see the asteroid with his telescope when the asteroid is within 145 million miles of Earth. 12 a. Write an expression that gives the distance of the asteroid from Earth as a function of x. b. For what values of x will the asteroid be in range of Carl s telescope? x a 1 Chapter 6 49 Glencoe Algebra 2

6 6-7 Enrichment Truth Tables In mathematics, the basic operations are addition, subtraction, multiplication, division, finding a root, and raising to a power. In logic, the basic operations are the following: not ( ), and ( ), or ( ), and implies ( ). If p and q are statements, then p means not p; q means not q; p q means p and q; p q means p or q; and p q means p implies q. The operations are defined by truth tables. On the left below is the truth table for the statement p. Notice that there are two possible conditions for p, true (T) or false (F). If p is true, p is false; if p is false, p is true. Also shown are the truth tables for p q, p q, and p q. p p p q p q p q p q p q p q T F F T T T T T F F T T T T F T T T T T F F F T F F T T F T T F F F F F F F F T You can use this information to find out under what conditions a complex statement is true. Example Under what conditions is p q true? Create the truth table for the statement. Use the information from the truth table above for p q to complete the last column. p q p p q T T F T T F F F F T T T F F T T The truth table indicates that p q is true in all cases except where p is true and q is false. Use truth tables to determine the conditions under which each statement is true. 1. p q 2. p (p q). (p q) ( p q) 4. (p q) (q p) 5. (p q) (q p) 6. ( p q) (p q) Chapter 6 50 Glencoe Algebra 2