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The best way is to solve it by graphing.
4x - 3y - 8 < 0 (1)
First graph the linear function: f(x,y) = 4x - 3y - 8 = 0.
When x = 0 ---> y = -8/3
When y = 0 ---> x = 2.
The graph is the line passing at 2 points A(0, -8/3) and B(2, 0).
Now use the check point method to find the area that is the solution set.
Check with the origin O. Substitute x = 0 and y = 0 into the inequality (1). We get -8 < 0. It is true, then the area containing the origin is the solution set.
Re-check with the point (3, 0). Substitute y = 0 and x = 3 into the inequality (1). We get: 12 - 8 < 0. It is not true, then the other area is the answer.

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Start from the right side of the equation. Refer to List of Trig Identities
in book titled:"Solving trig equations and inequalities" (Amazon e-book 2010)
1 - cos x = 2 sin^2 x/2 (Identity 11B)
sin x = 2sin x/2.cos x/2 (Identity 10)
Quotient: (1 - cos x)/sin x = (2sin^2 x/2)/(2sin x/2.cos/x/2)
After simplification, the quotient becomes:
!1 - cos x)/sin x = (sin x/2) / (cos x/2) = tg x/2

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Solve x^2 + 8x - 9 = 0.
There are 2 Tips you should know when you meet this special type of quadratic equation. The standard form of a quadratic equation is: ax^2 + bx + c = 0.
Tip 1. When a + b + c = 0, one real root is x1 = 1 and the other x2 = c/a.
Example 1. Solve x^2 + 8x - 9 = 0. You see that a + b + c = 1 + 8 - 9 = 0. This is Tip 1. One real root is x1 = 1 and the other x2 = c/a = -9/1 = -9.
Example 2. Solve 7x^2 - 15x + 8 = 0. You find 7 - 15 + 8 = 0. This is Tip 1. The 2 real roots are: x1 = 1 and x2 = c/a = 8/7.
TIP 2. When a - b + c = 1, one real root x1 = -1 and the other x2 = -c/a.
Example 3. Solve 3x^2 - 5x - 8 = 0. You find 3 -(-5) - 8 = 0. This is Tip 2. Then one real root x1 = -1 and the other x2 = -c/a = 8/1 = 8.
Example 4. Solve 5x^2 + 17x + 12 = 0 . You find 5 -(17) + 12 = 0. Then x1 = -1 and x2 = -c/a = -12/5.
I advise you to remember these 2 Tips. It will save you a lot of time in solving quadratic equations.
Or, you can solve the given equation x^2 + 8x - 9 = 0 by the popular factoring method. Find 2 numbers whose product is -9 and whose sum is 8. Proceed: (-1, 9),(1, -9)OK. Factor the equation: x^2 + 8x - 9 = (x - 1)(x + 9) = 0.
Next, solve the 2 binomials for x.
x - 1 = 0 --> x = 1 and (x + 9) = 0 --> x = -9.

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Always solve a quadratic inequality in 4 steps.
Step 1. Bring it to standard form: f(x) = 2x^2 + 3x - 9 < 0.
Step 2. Solve f(x) = 0. You can use any method you prefer. I use the new Diagonal Sum method. Roots have opposite signs. There are 3 probable root-pairs:
(-1/2, 9/1),(-3/1, 3/2),(-3/2, 3/1). The diagonal sum of the second set is -3 = -b. The 2 real roots are -3 and 3/2.
Or, you can solve it by the factoring ac method (You Tube). Find 2 number that their product is ac = -18, and their sum is b = 3. Proceed: [(-1, 18)(1, -18)(-2, 9)(2, -9)(-3, 6), OK]. Replace in the equation f(x) = 0 the quantity 3x by two quantities -3x and 6x.
2x^2 + 3x - 9 = 2x^2 - 3x + 6x - 9 = 0.
= 2x(x + 3)- 3(x +3) = (x + 3)(2x - 3). Solve the 2 binomials:
x + 3 = 0 ---> x = -3
2x - 3 = 0 ---> x = 3/2
Step 3. Solve the inequality f(x) < 0. Use the number line and test point method. Plot the 2 real roots -3 and 3/2 on the number line. Use the origin O as test point. Substitute x = 0 into the inequality. You get -9 < 0. It is true, then the origin O is on the true segment (-3, 3/2).
Step 4. Express the answer (solution set) of the inequality in the form of an open interval (-3, 3/2). The 2 end points -3 and 3/2 are not included in the solution set.
If in the inequality, there is an additional (=) sign (lesser or equal to), then the solution set is a closed interval [-3, 3/2]. The 2 end points -3 and 3/2 are included in the solution set.

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Problem: cos x = 2/3 ; tan x < 0 ; Find exact values of sin x.
Proposed solution.
Suppose a calculator gives cos a1(deg.) = 2/3. On the trig unit circle, there are 2 arcs that have the same cos value (2/3). They are a1(deg.) and a2 = (360 - a1). The tan a2 is negative (< 0). The exact value of sin a2 is given by the trig identity:
sin^2 a2 = 1 - cos^2 a2 = 1 - 4/9 = 5/9.
There are 2 answers: sin a2 = 2.24/3 = 0.75 and sin a2 = -0.75, but only the negative value (-0.75) is the right answer.