Inequalities

Question 1

solve 2x^2 < x + 3

Answer 1

2x^2 - x - 3 < 0

(2x-3)(x+1)

-1 < x < 1.5

Question 2

How do you solve |x^2 - 4x| < 3?

Answer 2

graph |x^2 + 4x| and y=3

find the critical values using (x^2 + 4x) - 3=0 and
-(x^2 +4x)=3

mark values on the graph and evaluate

Question 3

how do you solve 7x/3x+1 < 4-x?

Answer 3

first sketch 4-x and 7x/3x+1

find intersections: 7x/3x+1= 4-x

evaluate graph to find answer

Question 4

|3x|+x =< 2?

Answer 4

|3x| =< 2- x

3x =2-x

|-3x| = 2 - x

Question 5

What must you be careful of finding intersections of equations involving moduli?

Answer 5

not to forget any roots as |x| = |-x|

Question 6

Three steps in solving inequalities…

Answer 6

1. find critical values
2. use a sketch to identify solutions
3. write down answers checking carefully for validity

Question 7

multiplying by a negative quantity

Answer 7

changes the sign

Question 8

best approach when a modulus is involved

Answer 8

sketch

Question 9

how can you make sketching easier?

Answer 9

by rearranging the inequality

Question 10

what changes < to > or vice versa

Answer 10

multiplying/dividing inequality by a negative

Question 11

finding critical values

Answer 11

rearrange expression
replace inequality sign by =
solve

Question 12

algebraic fractions

Answer 12

multiply by the denominator quantity squared to ensure this is positive
if it’s negative it will invert the inequality

Question 13

factorisation

Answer 13

it’s always best to factorise as much as possible