Exercise algebra.inequalities.quadratic
Description
solve a quadratic inequation
Derivations
1.
x^2+3*x-4 < 0
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2+3*x-4 < 0)]x^2+3*x-4 == 0
⇒ algebra.equations.quadratic.nice-factors(x-1)*(x+4) == 0
⇒ algebra.equations.quadratic.product-zerox == 1 or x == -4
⇒ algebra.inequalities.give-solution, clipboard=[]-4 < x < 1
2.
x^2-4*x-12 > 0
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2-4*x-12 > 0)]x^2-4*x-12 == 0
⇒ algebra.equations.quadratic.nice-factors(x+2)*(x-6) == 0
⇒ algebra.equations.quadratic.product-zerox == -2 or x == 6
⇒ algebra.inequalities.give-solution, clipboard=[]x < -2 or x > 6
3.
-x^2-4*x+5 < 0
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (-x^2-4*x+5 < 0)]-x^2-4*x+5 == 0
⇒ algebra.equations.quadratic.simpler-polyx^2+4*x-5 == 0
⇒ algebra.equations.quadratic.nice-factors(x-1)*(x+5) == 0
⇒ algebra.equations.quadratic.product-zerox == 1 or x == -5
⇒ algebra.inequalities.give-solution, clipboard=[]x < -5 or x > 1
4.
-x^2+3*x+18 > 0
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (-x^2+3*x+18 > 0)]-x^2+3*x+18 == 0
⇒ algebra.equations.quadratic.simpler-polyx^2-3*x-18 == 0
⇒ algebra.equations.quadratic.nice-factors(x+3)*(x-6) == 0
⇒ algebra.equations.quadratic.product-zerox == -3 or x == 6
⇒ algebra.inequalities.give-solution, clipboard=[]-3 < x < 6
5.
1/2*x^2-3*x-8 < 0
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (1/2*x^2-3*x-8 < 0)]1/2*x^2-3*x-8 == 0
⇒ algebra.equations.quadratic.simpler-polyx^2-6*x-16 == 0
⇒ algebra.equations.quadratic.nice-factors(x+2)*(x-8) == 0
⇒ algebra.equations.quadratic.product-zerox == -2 or x == 8
⇒ algebra.inequalities.give-solution, clipboard=[]-2 < x < 8
6.
-2*x^2+10*x > 0
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (-2*x^2+10*x > 0)]-2*x^2+10*x == 0
⇒ algebra.equations.quadratic.common-factor-2*x*(x-5) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or x == 5
⇒ algebra.inequalities.give-solution, clipboard=[]0 < x < 5
7.
x^2+9*x < 3*x-5
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2+9*x < 3*x-5)]x^2+9*x == 3*x-5
⇒ algebra.equations.quadratic.move-leftx^2+6*x+5 == 0
⇒ algebra.equations.quadratic.nice-factors(x+1)*(x+5) == 0
⇒ algebra.equations.quadratic.product-zerox == -1 or x == -5
⇒ algebra.inequalities.give-solution, clipboard=[]-5 < x < -1
8.
x^2-x > 12
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2-x > 12)]x^2-x == 12
⇒ algebra.equations.quadratic.move-leftx^2-x-12 == 0
⇒ algebra.equations.quadratic.nice-factors(x+3)*(x-4) == 0
⇒ algebra.equations.quadratic.product-zerox == -3 or x == 4
⇒ algebra.inequalities.give-solution, clipboard=[]x < -3 or x > 4
9.
x^2-9/2*x < 7-3*x
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2-9/2*x < 7-3*x)]x^2-9/2*x == 7-3*x
⇒ algebra.equations.quadratic.move-leftx^2-3/2*x-7 == 0
⇒ algebra.equations.quadratic.simpler-poly2*x^2-3*x-14 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 121, a == 2, b == -3, c == -14, ineq == (x^2-9/2*x < 7-3*x)]x == 7/2 or x == -2
⇒ algebra.inequalities.give-solution, clipboard=[D == 121, a == 2, b == -3, c == -14]-2 < x < 7/2
10.
2*x^2-10*x > x^2-3*x
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (2*x^2-10*x > x^2-3*x)]2*x^2-10*x == x^2-3*x
⇒ algebra.equations.quadratic.move-leftx^2-7*x == 0
⇒ algebra.equations.quadratic.common-factorx*(x-7) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or x == 7
⇒ algebra.inequalities.give-solution, clipboard=[]x < 0 or x > 7
11.
4*x^2+6*x < x^2+3*x+18
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (4*x^2+6*x < x^2+3*x+18)]4*x^2+6*x == x^2+3*x+18
⇒ algebra.equations.quadratic.move-left3*x^2+3*x-18 == 0
⇒ algebra.equations.quadratic.simpler-polyx^2+x-6 == 0
⇒ algebra.equations.quadratic.nice-factors(x-2)*(x+3) == 0
⇒ algebra.equations.quadratic.product-zerox == 2 or x == -3
⇒ algebra.inequalities.give-solution, clipboard=[]-3 < x < 2
12.
2*x^2+6*x-10 > x^2+2*x-5
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (2*x^2+6*x-10 > x^2+2*x-5)]2*x^2+6*x-10 == x^2+2*x-5
⇒ algebra.equations.quadratic.move-leftx^2+4*x-5 == 0
⇒ algebra.equations.quadratic.nice-factors(x-1)*(x+5) == 0
⇒ algebra.equations.quadratic.product-zerox == 1 or x == -5
⇒ algebra.inequalities.give-solution, clipboard=[]x < -5 or x > 1
13.
x^2+9*x < 3*x-5
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2+9*x < 3*x-5)]x^2+9*x == 3*x-5
⇒ algebra.equations.quadratic.move-leftx^2+6*x+5 == 0
⇒ algebra.equations.quadratic.nice-factors(x+1)*(x+5) == 0
⇒ algebra.equations.quadratic.product-zerox == -1 or x == -5
⇒ algebra.inequalities.give-solution, clipboard=[]-5 < x < -1
14.
x^2-x > 12
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2-x > 12)]x^2-x == 12
⇒ algebra.equations.quadratic.move-leftx^2-x-12 == 0
⇒ algebra.equations.quadratic.nice-factors(x+3)*(x-4) == 0
⇒ algebra.equations.quadratic.product-zerox == -3 or x == 4
⇒ algebra.inequalities.give-solution, clipboard=[]x < -3 or x > 4
15.
x^2-9/2*x < 7-3*x
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2-9/2*x < 7-3*x)]x^2-9/2*x == 7-3*x
⇒ algebra.equations.quadratic.move-leftx^2-3/2*x-7 == 0
⇒ algebra.equations.quadratic.simpler-poly2*x^2-3*x-14 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == 121, a == 2, b == -3, c == -14, ineq == (x^2-9/2*x < 7-3*x)]x == 7/2 or x == -2
⇒ algebra.inequalities.give-solution, clipboard=[D == 121, a == 2, b == -3, c == -14]-2 < x < 7/2
16.
2*x^2-10*x > x^2-3*x
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (2*x^2-10*x > x^2-3*x)]2*x^2-10*x == x^2-3*x
⇒ algebra.equations.quadratic.move-leftx^2-7*x == 0
⇒ algebra.equations.quadratic.common-factorx*(x-7) == 0
⇒ algebra.equations.quadratic.product-zerox == 0 or x == 7
⇒ algebra.inequalities.give-solution, clipboard=[]x < 0 or x > 7
17.
4*x^2+6*x < x^2+3*x+18
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (4*x^2+6*x < x^2+3*x+18)]4*x^2+6*x == x^2+3*x+18
⇒ algebra.equations.quadratic.move-left3*x^2+3*x-18 == 0
⇒ algebra.equations.quadratic.simpler-polyx^2+x-6 == 0
⇒ algebra.equations.quadratic.nice-factors(x-2)*(x+3) == 0
⇒ algebra.equations.quadratic.product-zerox == 2 or x == -3
⇒ algebra.inequalities.give-solution, clipboard=[]-3 < x < 2
18.
2*x^2+6*x-10 > x^2+2*x-5
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (2*x^2+6*x-10 > x^2+2*x-5)]2*x^2+6*x-10 == x^2+2*x-5
⇒ algebra.equations.quadratic.move-leftx^2+4*x-5 == 0
⇒ algebra.equations.quadratic.nice-factors(x-1)*(x+5) == 0
⇒ algebra.equations.quadratic.product-zerox == 1 or x == -5
⇒ algebra.inequalities.give-solution, clipboard=[]x < -5 or x > 1
19.
x^2-x-7 > -100
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2-x-7 > -100)]x^2-x-7 == -100
⇒ algebra.equations.quadratic.move-leftx^2-x+93 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == -371, a == 1, b == -1, c == 93, ineq == (x^2-x-7 > -100)]false
⇒ algebra.inequalities.give-solution, clipboard=[D == -371, a == 1, b == -1, c == 93]true
20.
x^2-x-7 < -100
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2-x-7 < -100)]x^2-x-7 == -100
⇒ algebra.equations.quadratic.move-leftx^2-x+93 == 0
⇒ algebra.equations.quadratic.abc, clipboard=[D == -371, a == 1, b == -1, c == 93, ineq == (x^2-x-7 < -100)]false
⇒ algebra.inequalities.give-solution, clipboard=[D == -371, a == 1, b == -1, c == 93]false
21.
22.
23.
x^2 >= 0
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2 >= 0)]x^2 == 0
⇒ algebra.equations.coverup.powerx == 0
⇒ algebra.inequalities.give-solution, clipboard=[]true
24.
x^2 > 0
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2 > 0)]x^2 == 0
⇒ algebra.equations.coverup.powerx == 0
⇒ algebra.inequalities.give-solution, clipboard=[]x < 0 or x > 0
25.
x^2 < 0
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2 < 0)]x^2 == 0
⇒ algebra.equations.coverup.powerx == 0
⇒ algebra.inequalities.give-solution, clipboard=[]false
26.
x^2 <= 0
⇒ algebra.inequalities.to-equation, clipboard=[ineq == (x^2 <= 0)]x^2 == 0
⇒ algebra.equations.coverup.powerx == 0
⇒ algebra.inequalities.give-solution, clipboard=[]x == 0