Question
Solve the inequality
Solve the inequality by testing the values in the interval
Solve the inequality by separating into cases
Solve for x
x∈(−∞,32]∪[1,+∞)
Evaluate
3x2≥5x−2
Move the expression to the left side
3x2−(5x−2)≥0
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
3x2−5x+2≥0
Rewrite the expression
3x2−5x+2=0
Factor the expression
More Steps

Evaluate
3x2−5x+2
Rewrite the expression
3x2+(−2−3)x+2
Calculate
3x2−2x−3x+2
Rewrite the expression
x×3x−x×2−3x+2
Factor out x from the expression
x(3x−2)−3x+2
Factor out −1 from the expression
x(3x−2)−(3x−2)
Factor out 3x−2 from the expression
(x−1)(3x−2)
(x−1)(3x−2)=0
When the product of factors equals 0,at least one factor is 0
x−1=03x−2=0
Solve the equation for x
More Steps

Evaluate
x−1=0
Move the constant to the right-hand side and change its sign
x=0+1
Removing 0 doesn't change the value,so remove it from the expression
x=1
x=13x−2=0
Solve the equation for x
More Steps

Evaluate
3x−2=0
Move the constant to the right-hand side and change its sign
3x=0+2
Removing 0 doesn't change the value,so remove it from the expression
3x=2
Divide both sides
33x=32
Divide the numbers
x=32
x=1x=32
Determine the test intervals using the critical values
x<3232<x<1x>1
Choose a value form each interval
x1=−1x2=65x3=2
To determine if x<32 is the solution to the inequality,test if the chosen value x=−1 satisfies the initial inequality
More Steps

Evaluate
3(−1)2≥5(−1)−2
Simplify
More Steps

Evaluate
3(−1)2
Evaluate the power
3×1
Any expression multiplied by 1 remains the same
3
3≥5(−1)−2
Simplify
More Steps

Evaluate
5(−1)−2
Simplify
−5−2
Subtract the numbers
−7
3≥−7
Check the inequality
true
x<32 is the solutionx2=65x3=2
To determine if 32<x<1 is the solution to the inequality,test if the chosen value x=65 satisfies the initial inequality
More Steps

Evaluate
3(65)2≥5×65−2
Multiply the terms
More Steps

Evaluate
3(65)2
Evaluate the power
3×3625
Multiply the numbers
1225
1225≥5×65−2
Simplify
More Steps

Evaluate
5×65−2
Multiply the numbers
625−2
Reduce fractions to a common denominator
625−62×6
Write all numerators above the common denominator
625−2×6
Multiply the numbers
625−12
Subtract the numbers
613
1225≥613
Calculate
2.083˙≥613
Calculate
2.083˙≥2.16˙
Check the inequality
false
x<32 is the solution32<x<1 is not a solutionx3=2
To determine if x>1 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
More Steps

Evaluate
3×22≥5×2−2
Multiply the terms
More Steps

Evaluate
3×22
Evaluate the power
3×4
Multiply the numbers
12
12≥5×2−2
Simplify
More Steps

Evaluate
5×2−2
Multiply the numbers
10−2
Subtract the numbers
8
12≥8
Check the inequality
true
x<32 is the solution32<x<1 is not a solutionx>1 is the solution
The original inequality is a nonstrict inequality,so include the critical value in the solution
x≤32 is the solutionx≥1 is the solution
Solution
x∈(−∞,32]∪[1,+∞)
Show Solution
