Quadratic inequalities

Question 1

Example 1 - Solve the inequality

2(3x-5)≥43

Answer 1

2(3x-5)≥43
6x-10≥43
6x≥53
x≥53/6

Question 2

Example 2- Solve the inequality

2x+9>19-8x

Answer 2

2x+9>19-8x
10x+9>19
10x>10
x>1

Question 3

Example 3 - x is an integer. Write down all the solutions of the inequality.
3<2x+1<13

Answer 3

3<2x+1<13
2<2x<12
1

Question 4

What is regular notation?

Answer 4

e.g,x <0 or x>3

Question 5

What is set notation?

Answer 5

E.g. {x:x<0 or x>3}

Question 6

What are the steps to solving quadratic inequalities?

Answer 6

1. Solve the quadratic
2. Sketch the graph
3. Shade the parts you need (beard or sideburns)
4. Write the inequality

Question 7

Example 1 - Solve the following inequality:

(x-1)(x+2)<0

Answer 7

(x-1)(x+2)<0
(x-1)(x+2)=0
x-1=0
x=1
x+2=0
x=-2
x=1, x=-2
When you sketch the graph, you are drawing the points at which it crosses the x axis when y=0. So this is -2 and 1, both of which are at the x axis. It should be a u shaped graph. Also, you should have shaded it like a beard, as they are part of the same answer. As x is between -2 and 1, then you're shading all the parts on the graph between -2 and 1. 
-2

Question 8

Example 2- solve the inequality:

2x^2-7x+3≥0

Answer 8

2x^2-7x+3≥0
Ac method:
x to give ac:6
\+to give b:-7
-6, -1
2x^2-6x-1x+3=0
First half:
2x(x-3)
Second half:
-1(x-3)
(2x-1)(x-3)=0
2x-1=0
2x=1
x=1/2
x-3=0
x=3
Sketch: mark the points at which it crosses the x axis (1/2 and 3). If we look back to the original question, it uses this symbol ≥0. It says everything is bigger or equal to zero. Because of this, then we know our graph must be shaded in with sideburns, as we are looking for every number thats y value is above zero, and sideburns shade in the parts above not below.
x≤0.5 or x≥3. We know to use these symbols as when we look at the graph, we have shaded in the parts that go before 0.5 and after 3

Question 9

Example3 - Solve 2-x-x^2≥0

Answer 9

2-x-x^2≥0
Solve: 2-x-x^2=0
-x^2-x+2=0
x^2+x-2=0 (got this from multiplying by -1)
shorter ac method
x to give ac:-2
\+ to give b: 1
2, -1
(x+2)(x-1)=0
x+2=0
x=-2
x-1=0
x=1
When you draw the graph, you need to draw an n shape as it is -x^2. Then, label the x axis with -2 and 1. Also, as it says at the start using this symbol ≥ everything is equal to or bigger than zero, then we know we are looking for values that go above y=0 (that have a positive y value), so we shade in a beard as that is how to shade in that have a positive y value.
-2≤x≤1
We know this because, looking at where we have shaded on the graph, x is between -2 and 1. It is only one answer as well as it is a beard not sideburns