Linear vs. Quadratic
True or False: The inequality (3x - 5 >x² ) is a quadratic inequality
True (It can be rearranged into the standard form (x² - 3x + 5 < 0)
Find the two boundary numbers for the inequality x² - 9 ≤ 0.
3 and -3 (since 3 × 3 = 9 and -3 × -3 = 9).
If an inequality uses the symbol < or >, do you use open circles or solid closed dots on the boundary numbers?
Open circles
What is the solution to x² < -5?
No solution
What is the highest exponent (degree) in any quadratic inequality?
2
What are the two boundary numbers for (x - 2)(x - 5) > 0
2 and 5
If an inequality uses the symbol ≤ or ≥, do you use open circles or solid closed dots?
Solid closed dots
What is the solution to x² ≥ 0?
All real numbers
Identify which of the following is not a quadratic inequality:
A) (x(x - 2)≤4)
B) (x² + 3x^3 > 0)
C) (x²≥9)
Answer: B. (It contains an (x^3) term, making it a cubic inequality).
If you factor x² - 4x = 0 into x(x - 4) = 0, what are your two boundary numbers?
0 and 4
Look at this solution: [3, 7]. Does this interval notation mean the numbers between 3 and 7 are shaded, or the numbers outside of them?
Between 3 and 7
A square garden has a side length of x. Write an inequality showing that the area of the garden must be greater than 50 square feet.
x² > 50
True or False: The inequality (1/x² + 3x > 2)is a quadratic inequality
False. (Because the (x²) is in the denominator, this is a rational inequality, not a quadratic polynomial).
Find the boundary numbers for x² - 7x + 10 = 0
2 and 5, factored as (x-2)(x-5)
If a graph shows two arrows pointing outward away from each other starting at -2 and 2, is x = 0 a solution?
No (0 is in the unshaded middle section).
A rectangle's area is given by x(x + 2). Set up an inequality showing that this area is at most 24 square units.
x(x + 2) ≤ 24 (or x² + 2x ≤ 24).
For what specific value of (k) does the inequality (k - 3)x² + 5x - 2 > 0 fail to be a quadratic inequality?
(k = 3). (If (k = 3), the x² coefficient becomes 3 - 3 = 0, which completely eliminates the quadratic term and turns it into the linear inequality (5x - 2 > 0).
If you try to find the boundary numbers for x² + 10 < 0, why will you find no real numbers?
A squared number can never be negative, so it can never equal -10.
Match the graph description to the inequality: The boundary points are open circles at -1 and 4, and the shading is in the middle. Does this match x² - 3x - 4 < 0 or x² - 3x - 4 > 0?
x² - 3x - 4 < 0 (The "<" sign shades the negative, inside region)
If a parabola sits entirely above the x-axis and never touches it, what is the solution to the inequality ax² + bx + c < 0?
No solution (it is never below the axis).