Grade 9
Lesson 49: Solving polynomial inequalities
In this section we will be solving (single) inequalities that involve polynomials of degree at least two. Since it’s easier to see the process as we work an example let’s do that. As with the linear inequalities, we are looking for all the values of the variable that will make the inequality true. This means that our solution will almost certainly involve inequalities as well. The process that we’re going to go through will give the answers in that form.
How To Solve Polynomial Inequalities?
Here are the steps to solve polynomial inequalities:
 Make your inequality looks like P(x) < 0 (the inequality symbol can also be >, ≤ or ≥)
 Find the zeros of P(x) (by factoring it)
 Plot the zeros on the number line
 Pick a test point on each interval and plugin it to P(x)
 If after plugin you get it true then the interval is a part of the solution otherwise it’s not
The following examples will guide you step by step to solve the polynomial inequalities:
Example 1: Solve x² – 10 < 3x
step1: Make your inequality looks like P(x) < 0 (the inequality symbol can also be >, ≤ or ≥)
x² – 3x 10 < 0
step2: Find the zeros of P(x) (by factoring it)
(x + 2)(x – 5) < 0
So, the zeros of x are {2 , 5}
step3: Plot the zeros on the number line
step4: Pick a test point on each interval and plugin it to P(x)
step5: If after plugin you get it true then the interval is a part of the solution otherwise it’s not
So the solution is between 2 and 5, 2 < x < 5
Example 2: Solve x²( x² + 4x – 12) ≤ 0
step1: Make your inequality looks like P(x) < 0 (the inequality symbol can also be >, ≤ or ≥)
x²( x² + 4x – 12) ≤ 0
step2: Find the zeros of P(x) (by factoring it)
x²(x + 6)(x – 2) ≤ 0
So, the zeros of x are {6, 0 , 2}
step3: Plot the zeros on the number line
step4: Pick a test point on each interval and plugin it to P(x)
step5: If after plugin you get it true then the interval is a part of the solution otherwise it’s not
So the solution is between 6 and 2, 6 ≤ x ≤ 2
Practice 74: Trigonometry
Answer the following questions

Random Trigonometry
Lots of random trigonometry problems
Back to Lesson 74: Trgonometry
Lesson 74: Trigonometry
Trigonometry (from Greek trigōnon “triangle” + metron “measure”) is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides.
Sine, Cosine and Tangent
The three main functions in trigonometry are Sine, Cosine and Tangent. They are often shortened to sin, cos and tan. they are based on a RightAngled Triangle, where they help us finding an unknown angle or side in a rightangled triangle.
Before getting stuck into these functions, it helps to give a name to each side of a right triangle:
 “Opposite” is the side that faces angle θ
 “Adjacent” is the side (next to) angle θ
 “Hypotenuse” is the longest side and it always faces the right angle
For a right triangle with an angle θ, the functions Sine, Cosine and Tangent are calculated this way:
How to Remember them?
well “Sohcahtoa” may be easy for you to remember…
Soh…

Sine = Opposite / Hypotenuse

…cah…

Cosine = Adjacent / Hypotenuse

…toa

Tangent = Opposite / Adjacent

Why are these functions important?
 Because they let you work out angles when you know sides
 And they let you work out sides when you know angles
Example: A 5ft ladder leans against a wall as shown.
What is the angle between the ladder and the wall?
The answer can be found using: Sine, Cosine or Tangent!
But which one to use?
Remember: we have a special phrase “SOHCAHTOA” to help us, and we use it like this:
Step 1: find the names of the two sides you know
In our ladder example we know the length of:
 the side Opposite the angle “x” (2.5 ft)
 the long side, called the “Hypotenuse” (5 ft)
Step 2: now use the first letters of those two sides (Opposite and Hypotenuse) and the phrase “SOHCAHTOA” to find which one of Sine, Cosine or Tangent to use:
SOH…

Sine: sin(θ) = Opposite / Hypotenuse

…CAH…

Cosine: cos(θ) = Adjacent / Hypotenuse

…TOA

Tangent: tan(θ) = Opposite / Adjacent

In our example that is Opposite and Hypotenuse, and that gives us “SOHcahtoa”, which tells us we need to use Sine.
Step 3: Put our values into the Sine equation:
Sin (x) = Opposite / Hypotenuse = 2.5 / 5 = 0.5
Step 4: Now solve that equation!
sin (x) = 0.5
Next (trust me for the moment) we can rearrange that into this:
x = sin^{1} (0.5)
And then get our calculator, key in 0.5 and use the sin^{1} button to get the answer:
x = 30°
But what is the meaning of sin^{1} … ?
Well, the Sine function “sin” takes an angle and gives us the ratio “opposite/hypotenuse”,
But in this case we know the ratio “opposite/hypotenuse” but want to know the angle.
So we want to go backwards. That is why we we use sin^{1}, which means “inverse sine”.
Example:
 Sine Function: sin(30°) = 0.5
 Inverse Sine Function: sin^{1}(0.5) = 30°
Examples:
Let’s look at a couple more examples:
1 ) Find the size of the angle of elevation (x) of the plane from point A on the ground.
Step 1 The two sides we know are Opposite (300) and Adjacent (400).
Step 2 SOHCAHTOA tells us we must use Tangent.
Step 3 Use your calculator to calculate Opposite/Adjacent
Step 4 Find the angle from your calculator using tan^{1}
Tan x° = opposite/adjacent
Tan x° = 300/400
Tan x° = 0.75
x° = tan^{1} (0.75)
x° = 36.9°
2 ) Find the size of angle a°
Step 1 The two sides we know are Adjacent (6,750) and Hypotenuse (8,100).
Step 2 SOHCAHTOA tells us we must use Cosine.
Step 3 Use your calculator to calculate Adjacent / Hypotenuse
Step 4 Find the angle from your calculator using cos^{1}
cos a° = 6,750/8,100
cos a° = 0.8333
Practice 81: Angles of Polygons
Answer the following questions






Back to Lesson 81: angles of polygons
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