_________ \/ x + 8 = x + 2The general method to solve an equation is to replace, in succession, the equation by an equivalent equation.
We tend to square both sides of the equation, but the next equivalence is false.
_________ \/ x + 8 = x + 2 <=> x + 8 = (x + 2)^{2} Indeed, -4 is solution of x + 8 = (x + 2)^{2} but it is not a solution of _________ \/ x + 8 = x + 2 It is obvious that the following expression is correct. _________ \/ x + 8 = x + 2 => x + 8 = (x + 2)^{2}All solutions of the equation on the left side are solutions of the equation on the right side but not vice versa. The equation on the right can have more solutions.
Nevertheless, we will solve irrational equations by squaring both sides.
But we must be aware that, by
squaring, the new equation can have more solutions than the original one.
At the end, the 'false solutions' are deleted.
There are different ways to find these 'false solutions'.
One way is to build suitable inequalities in advance, to find the 'false solutions'.
We don't follow this method here.
The simplest way is to test each solution to the given equation. If the given equation
is not satisfied for that solution, it is labeled as a 'false solution'.
Let's use this procedure to the equation
_________ \/ x + 8 = x + 2 => x + 8 = (x + 2)^{2} <=> ... <=> x = 1 of x = -4We test these two values to the given equation. We see that -4 is a false solution and it must be deleted. The only solution is 1.
_______ 1 + \/ x^{2} -9 = x _______ <=> \/ x^{2} -9 = x - 1 => x^{2} -9 = (x - 1)^{2}After developing and simplifying we find x = 5.
_______ _______ \/ 2x + 8 + \/ x + 5 = 7 _______ _______ <=> \/ 2x + 8 = 7 - \/ x + 5 _______ => 2x + 8 = (7 - \/ x + 5 )^{2} After developing and simplifying _______ x - 46 = -14 \/ x + 5 Squaring again, we obtain => x^{2} - 288 x + 1136 = 0 with solutions 4 and 248. 4 is the only solution of the original equation.
_______ _______ _______ \/ x + 3 + \/ x + 8 = \/ x + 24 We square each side and then we simplify ______________ => 2 \/ (x + 3)(x + 8) = 13 - x Squaring again, we obtain => 3x^{2} + 70x - 73 = 0 x = 1 is the only solution of the original equation!
______________ ______________ / _______ / _______ \/ x + \/ x+ 11 + \/ x - \/ x+ 11 = 4 We square each side and then we simplify _____________ => x + \/ x^{2} - x - 11 = 8 We bring x to rhe right side and then we square again. We find x = 5 as a solution and this solution is not false.
3 _______ \/ 2x - 5 = 3 By presence of the cube root, we can write 3 _______ \/ 2x - 5 = 3 <=> 2 x - 5 = 27 <=> x = 16
________ _______ \/ cos(2x) - \/ cos(x) = 0 ________ _______ <=> \/ cos(2x) = \/ cos(x) => cos(2x) = cos(x) <=> 2x = x + 2 k pi or 2x = - x + 2 k pi <=> x = 2 k pi or x = 2 k pi/3 <=> x = 2 k pi or x = 2 pi/3 + 2 k pi or x = 4 pi/3 + 2 k pi x = 2 pi/3 + 2 k pi and x = 4 pi/3 + 2 k pi are false solutions The only solutions are x = 2 k pi
_______ 3 + \/ 3x + 1 = x ( Solution : 8) _______ _______ \/ x + 27 - \/ x - 5 = 2 ( Solution : 54) _______ _______ \/ 7x + 2 - \/ 3x + 1 = 1 ( Solution : 1) __________ / ______ _______ \/ 2 \/ x + 1 = \/ 3x - 5 ( Solution : 3)
Bring all terms of the inequality to the left side. This creates an irrational function f(x) in the left side of the inequality. Find the domain of the function f(x).
step 2
Find the zero's of the irrational function f(x).
step 3
Draw the axis of the real numbers.
Rule out the area that does not belong to the domain of f(x)
Mark the zero's of f(x) on the axis.
Determine the sign of f(x) in all intermediate intervals. This can be done
with the aid of the image of a simple x-value.
step 4
From the result of step 3, you can read the solution set of the inequality.
x + 8 > x + 2 Step 1: _________ \/ x + 8 - x - 2 > 0 The domain of the function f(x) on the left side is [-8, +infty) Step 2: Solve the equation _________ \/ x + 8 - x - 2 = 0 The solution is 1. Step 3: Determine the sign of f(x) x | -8 1 ---|----------------------------- f(x)| //////////// + + + + 0 - - - Step 4: The solution set of the inequality is [-8,1)
_______ _______ \/ 2x + 8 < 7 - \/ x + 5 We use the four step method _______ _______ \/ 2x + 8 + \/ x + 5 - 7 < 0 The domain of the function f(x) in the left side is [-4, +infty) The set of zero's is {4} Sign investigation: x | -4 4 ---|----------------------------- f(x)| //////////// - - - - 0 + + + The solution set of the inequality is [-4, 4)
3 _______ \/ 2x - 5 < 3 We use the four step method 3 _______ \/ 2x - 5 - 3 < 0 The domain of the function f(x) in the left side is R The only zero of f(x) is 16 Sign investigation: x | 16 ---|------------------ f(x)| - - - - 0 + + + The solution set of the inequality is (-infty, 16)