Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x≥2310
Alternative Form
x∈[2310,+∞)
Evaluate
x−3<x2×4x−5
Find the domain
More Steps
Evaluate
x2×4x−5≥0
Multiply
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Evaluate
x2×4x
Multiply the terms with the same base by adding their exponents
x2+1×4
Add the numbers
x3×4
Use the commutative property to reorder the terms
4x3
4x3−5≥0
Move the constant to the right side
4x3≥5
Divide both sides
44x3≥45
Divide the numbers
x3≥45
Take the 3-th root on both sides of the equation
3x3≥345
Calculate
x≥345
Simplify the root
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Evaluate
345
To take a root of a fraction,take the root of the numerator and denominator separately
3435
Multiply by the Conjugate
34×34235×342
Simplify
34×34235×232
Multiply the numbers
34×3422310
Multiply the numbers
222310
Reduce the fraction
2310
x≥2310
x−3<x2×4x−5,x≥2310
Multiply
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Evaluate
x2×4x
Multiply the terms with the same base by adding their exponents
x2+1×4
Add the numbers
x3×4
Use the commutative property to reorder the terms
4x3
x−3<4x3−5
Swap the sides
4x3−5>x−3
Separate the inequality into 2 possible cases
4x3−5>x−3,x−3≥04x3−5>x−3,x−3<0
Solve the inequality
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Solve the inequality
4x3−5>x−3
Square both sides of the inequality
4x3−5>(x−3)2
Expand the expression
4x3−5>x2−6x+9
Move the expression to the left side
4x3−5−(x2−6x+9)>0
Subtract the terms
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Evaluate
4x3−5−(x2−6x+9)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
4x3−5−x2+6x−9
Subtract the numbers
4x3−14−x2+6x
4x3−14−x2+6x>0
Rewrite the expression
4x3−14−x2+6x=0
Find the critical values by solving the corresponding equation
x≈1.261157
Determine the test intervals using the critical values
x<1.261157x>1.261157
Choose a value form each interval
x1=0x2=2
To determine if x<1.261157 is the solution to the inequality,test if the chosen value x=0 satisfies the initial inequality
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Evaluate
4×03−14−02+6×0>0
Any expression multiplied by 0 equals 0
4×03−14−02+0>0
Simplify
−14>0
Check the inequality
false
x<1.261157 is not a solutionx2=2
To determine if x>1.261157 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
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Evaluate
4×23−14−22+6×2>0
Simplify
26>0
Check the inequality
true
x<1.261157 is not a solutionx>1.261157 is the solution
The original inequality is a strict inequality,so does not include the critical value ,the final solution is x>1.261157
x>1.261157
x>1.261157,x−3≥04x3−5>x−3,x−3<0
Solve the inequality
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Evaluate
x−3≥0
Move the constant to the right side
x≥0+3
Removing 0 doesn't change the value,so remove it from the expression
x≥3
x>1.261157,x≥34x3−5>x−3,x−3<0
Since the left-hand side is always positive or 0,and the right-hand side is always negative,the statement is true for any value of x
x>1.261157,x≥3x∈R,x−3<0
Solve the inequality
More Steps
Evaluate
x−3<0
Move the constant to the right side
x<0+3
Removing 0 doesn't change the value,so remove it from the expression
x<3
x>1.261157,x≥3x∈R,x<3
Find the intersection
x≥3x∈R,x<3
Find the intersection
x≥3x<3
Find the union
x∈R
Check if the solution is in the defined range
x∈R,x≥2310
Solution
x≥2310
Alternative Form
x∈[2310,+∞)
Show Solution
