1. √(x + 6) = -8
x=?
2. 4√(2x - 3) < 12
solve the inequality
3. 4 ≥ √(6x - 12) - 8
solve the inequality
4. √(3x - 5) = x - 3
x=?
5. √(4 - x) ≥ √(3x + 4)
solve the inequality
6. x + 3 = √(5x + 4) + 1
x=?
I will be happy to critique your work.
I will do #2 and #4, you can try the rest by following the same method.
#2.
4√(2x - 3) < 12
√(2x - 3) < 3 , now square both sides
2x - 3 < 9
2x < 12
x < 6
since we squared, we have to check our answer.
try x=2
LS = 4√(2(2)- 3) = 4
RS = 12
Is 4 < 12 ? Yes!
so x < 6
#4
√(3x - 5) = x - 3
square both sides
3x-5 = x^2 - 6x + 9
x^2 - 9x + 12 = 0
(x-7)(x-2) = 0
x = 7 or x = 2
checking:
if x=7
LS = √(21-5) = 4
RS = 7-3 = 4 , YES
if x=2
LS = √(4-1) = 1
RS = 2-3 = -1 , NO
so x = 7
Sure! I can help you solve these equations step-by-step.
1. √(x + 6) = -8
To start, square both sides of the equation to eliminate the square root:
(x + 6) = (-8)^2
x + 6 = 64
Next, isolate the variable x:
x = 64 - 6
x = 58
Therefore, the solution to the equation is x = 58.
2. 4√(2x - 3) < 12
First, divide both sides of the inequality by 4 to isolate the square root:
√(2x - 3) < 3
Next, square both sides of the inequality to eliminate the square root:
2x - 3 < 3^2
2x - 3 < 9
Now, add 3 to both sides of the inequality:
2x < 9 + 3
2x < 12
Finally, divide both sides of the inequality by 2 to solve for x:
x < 6
Therefore, the solution to the inequality is x < 6.
3. 4 ≥ √(6x - 12) - 8
First, add 8 to both sides of the inequality to isolate the square root:
4 + 8 ≥ √(6x - 12)
12 ≥ √(6x - 12)
Next, square both sides of the inequality to eliminate the square root:
12^2 ≥ (6x - 12)
144 ≥ 6x - 12
Now, add 12 to both sides of the inequality:
144 + 12 ≥ 6x
156 ≥ 6x
Finally, divide both sides of the inequality by 6 to solve for x:
26 ≥ x
Therefore, the solution to the inequality is x ≤ 26.
4. √(3x - 5) = x - 3
To start, square both sides of the equation to eliminate the square root:
3x - 5 = (x - 3)^2
3x - 5 = x^2 - 6x + 9
Now, rearrange the equation to form a quadratic equation:
x^2 - 9x + 14 = 0
Next, factor the quadratic equation:
(x - 2)(x - 7) = 0
Set each factor equal to zero and solve for x:
x - 2 = 0
x = 2
x - 7 = 0
x = 7
Therefore, the solutions to the equation are x = 2 and x = 7.
5. √(4 - x) ≥ √(3x + 4)
First, square both sides of the inequality to eliminate the square root:
4 - x ≥ 3x + 4
Now, subtract 4 from both sides of the inequality:
-x ≥ 3x
Next, subtract 3x from both sides of the inequality:
-4x ≥ 0
Finally, divide both sides of the inequality by -4. Remember to flip the inequality sign since we're dividing by a negative number:
x ≤ 0
Therefore, the solution to the inequality is x ≤ 0.
6. x + 3 = √(5x + 4) + 1
To start, square both sides of the equation to eliminate the square root:
(x + 3)^2 = (√(5x + 4) + 1)^2
(x + 3)(x + 3) = (5x + 4 + 1)^2
(x + 3)(x + 3) = (5x + 5)^2
Simplify both sides of the equation:
x^2 + 6x + 9=x^2 + 10x + 25
Now, rearrange the equation to form a quadratic equation:
x^2 + 6x + 9 - x^2 -10x - 25 = 0
Combine like terms:
-4x - 16 = 0
Now, add 16 to both sides of the equation:
-4x = 16
Finally, divide both sides of the equation by -4 to solve for x:
x = -4
Therefore, the solution to the equation is x = -4.
I hope this helps! Let me know if you have any other questions.
To solve these equations and inequalities involving square roots, follow these steps:
1. Isolate the square root term: Move any constant terms to the opposite side of the equation or inequality.
For example, in equation 1, we can start by isolating the square root term:
√(x + 6) = -8 becomes √(x + 6) + 8 = 0
2. Square both sides of the equation or inequality. This eliminates the square root term.
Using equation 1 as an example:
(√(x + 6) + 8)^2 = 0^2
Simplifying: (x + 6) + 16√(x + 6) + 64 = 0
3. Solve the resulting equation or inequality without the square root term.
Continue with example 1:
x + 6 + 16√(x + 6) + 64 = 0
Rearrange the equation: x + 70 + 16√(x + 6) = 0
Now, solve for x algebraically. In this case, the equation can be further simplified and solved:
16√(x + 6) = -x - 70
Squaring both sides once again: 256(x + 6) = (x + 70)^2
Expand and simplify: 256x + 1536 = x^2 + 140x + 4900
Rearrange the equation: x^2 - 116x + 3364 = 0
Factor or use the quadratic formula to obtain the two possible solutions for x.
4. Check the solutions: Substitute the obtained values of x back into the original equation or inequality to ensure they are valid solutions.
Repeat these steps for each given equation or inequality involving square roots.