Roots & Test Points
The critical points of the inequality x² - 5x + 6 ≤ 0.
2 and 3
For an upward-facing parabola, if the graph is below the x-axis, the quadratic is this type of value.
That is negative (less than zero)
What are the critical values (or roots) for the inequality x^2-9>0?
x=3 and x=-3
When graphing an inequality with ≤ or ≥ on a number line, do you use open or closed circles for the boundary points?
closed (solid) circles
Solve for x: x^2<16.
-4<x<4 or (-4,4)
To solve x² + 2x - 8 > 0, these are the three regions you must test.
x < -4, -4 < x < 2, and x > 2
If the vertex of y = x² - 4x + 3 is (2, -1), the graph is below the x-axis between these two x-values.
1 and 3
Rewrite x^2-5x≤-6 in standard quadratic inequality form (ax^2+bx+c≤0).
x^2-5x+6≤0
Express the inequality solution -3<x<5 in standard interval notation.
(-3,5)
Solve for x: (x-1)(x-5)≤0.
1≤x≤5 or [1,5]
The true region for the test point x = 0 in the inequality x² + x - 6 < 0.
(True)
The dashed or solid boundary line used for the inequality y > x² - 4.
That is a dashed line
Factoring gives (x-4)(x+2)<0. What are the two critical boundary numbers on the number line?
x=4 and x=-2
Express the solution x≤-1 or x≥7 in interval notation using the union symbol.
(-∞,-1]∪[7,∞)
Solve for x: x^2+3x>0.
x<-3 or x>0 interval notation as (-∞,-3)∪(0,∞)
The solution interval for the inequality x² - 3x - 10 ≤ 0.
[-2, 5]
The shaded region (inside or outside the parabola) for the inequality y ≤ x² - 9.
inside (and on) the parabola
What are the critical values for the inequality 2x^2+5x-3≥0?
x=1/2 and x=-3
If a quadratic inequality solution set covers every real number, how is this written in interval notation?
(-∞,∞)
Solve for x: x^2-6x+9>0.
all real numbers except x=3 OR (-∞,3)∪(3,∞)
A rocket's height is h(t)=-5t^2+30t+35 meters. During what time interval t is the rocket at or above 70 meters high?
between t=1.5 seconds and t=4.5 seconds (or [1.5,4.5]
If a<0 (parabola opens downward) and the vertex sits below the x-axis, what is the solution set for ax^2+bx+c>0?
No Solution (or the empty set, ∅)
What occurs when a quadratic inequality's underlying equation ax^2+bx+c=0 has a negative discriminant (b^2-4ac<0)?
"it has no real critical values" (or "the parabola does not cross the x-axis")
What symbol is used in interval notation when a boundary value is not included due to a strict inequality sign (< or >)?
( or )
Solve for x: x^2+4x+4≤0.
x=-2;(x+2)^2≤0 is only true when (x+2)^2=0, so x=-2