Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
Solve the inequality by testing the values in the interval
Solve the inequality by separating into cases
Solve for p
−2≤p≤32
Alternative Form
p∈[−2,32]
Evaluate
(p−2)2−4p2≥0
Rearrange the terms
−3p2−4p+4≥0
Rewrite the expression
−3p2−4p+4=0
Factor the expression
More Steps
Evaluate
−3p2−4p+4
Rewrite the expression
−3p2+(2−6)p+4
Calculate
−3p2+2p−6p+4
Rewrite the expression
−p×3p+p×2−2×3p+2×2
Factor out −p from the expression
−p(3p−2)−2×3p+2×2
Factor out −2 from the expression
−p(3p−2)−2(3p−2)
Factor out 3p−2 from the expression
(−p−2)(3p−2)
(−p−2)(3p−2)=0
When the product of factors equals 0,at least one factor is 0
−p−2=03p−2=0
Solve the equation for p
More Steps
Evaluate
−p−2=0
Move the constant to the right-hand side and change its sign
−p=0+2
Removing 0 doesn't change the value,so remove it from the expression
−p=2
Change the signs on both sides of the equation
p=−2
p=−23p−2=0
Solve the equation for p
More Steps
Evaluate
3p−2=0
Move the constant to the right-hand side and change its sign
3p=0+2
Removing 0 doesn't change the value,so remove it from the expression
3p=2
Divide both sides
33p=32
Divide the numbers
p=32
p=−2p=32
Determine the test intervals using the critical values
p<−2−2<p<32p>32
Choose a value form each interval
p1=−3p2=−1p3=2
To determine if p<−2 is the solution to the inequality,test if the chosen value p=−3 satisfies the initial inequality
More Steps
Evaluate
(−3−2)2−4(−3)2≥0
Simplify
More Steps
Evaluate
(−3−2)2−4(−3)2
Subtract the numbers
(−5)2−4(−3)2
Multiply the terms
(−5)2−36
Rewrite the expression
52−36
Evaluate the power
25−36
Subtract the numbers
−11
−11≥0
Check the inequality
false
p<−2 is not a solutionp2=−1p3=2
To determine if −2<p<32 is the solution to the inequality,test if the chosen value p=−1 satisfies the initial inequality
More Steps
Evaluate
(−1−2)2−4(−1)2≥0
Simplify
More Steps
Evaluate
(−1−2)2−4(−1)2
Subtract the numbers
(−3)2−4(−1)2
Evaluate the power
(−3)2−4×1
Any expression multiplied by 1 remains the same
(−3)2−4
Rewrite the expression
32−4
Evaluate the power
9−4
Subtract the numbers
5
5≥0
Check the inequality
true
p<−2 is not a solution−2<p<32 is the solutionp3=2
To determine if p>32 is the solution to the inequality,test if the chosen value p=2 satisfies the initial inequality
More Steps
Evaluate
(2−2)2−4×22≥0
Simplify
More Steps
Evaluate
(2−2)2−4×22
Subtract the numbers
02−4×22
Calculate
0−4×22
Multiply the terms
0−24
Removing 0 doesn't change the value,so remove it from the expression
−24
Evaluate the power
−16
−16≥0
Check the inequality
false
p<−2 is not a solution−2<p<32 is the solutionp>32 is not a solution
The original inequality is a nonstrict inequality,so include the critical value in the solution
−2≤p≤32 is the solution
Solution
−2≤p≤32
Alternative Form
p∈[−2,32]
Show Solution
