Algebra 7

Consider the linear equation
2x + 1 = 5
If we solve this equation we get x = 2. What happens if we plug x = 2 into the left hand side of the equation? We get 5. What happens if we plug a value of x bigger than 2 into the equation? We get an answer bigger than 5. And if we plug a value less than 2 into the equation we get an answer smaller than 5. So the equation or inequality
2x + 1 > 5
has the solution x > 2 and the inequality
2x + 1 < 5
has the solution x < 2.
Note that in some cases we may want to solve an equation like
2x + 1 >= 5
ie we want to allow solutions which equal 5 as well as those that are greater than 5. In this case we get x >= 2 as the solution. [It is difficult to show the exact symbol for greater than or equal to on this site. In practice the equal sign is put on the lower part of the greater than sign, giving a greater sign (the > ) with two lines at the bottom rather than one.]
We can solve inequalities in exactly the same way as we solve linear equations. Let`s look at the equation
5x + 4 >= 9.
If we treat this as a normal equation we would move the 4 over to the right hand side, changing it tipo minus 4, to get
5x >= 9 - 4
or
5x >= 5.
Now divide by 5 to get
x >= 1
which is the solution to the inequality. Any x value greater than or equal to one will solve the equation or inequality 5x + 4 >= 9.

Example 1 (Foundation Nov 14 Paper 1 Q17(b) )
Solve the inequality 9x + 5 < 77 (Answer x < 8)
Example 2 (Foundation May 15 Paper 2 Q14(c) )
Find all integer values of n that satisfy the inequality 5 <= 3n < 18 (Answer n=2,3,4 or 5)
Example 3 (Foundation Nov 15 Paper 2 Q15(c) )
Find all integer values of n that satisfy the inequality 6 < 2n < 13 (Answer n=4,5 or 6)
Example 4
Solve the inequality 3x + 2 >= 2x - 1 (Answer x >= -3)
Example 5
Solve the inequality 2x - 12 < 23 - 5x (Answer x < 5)
Example 6
Solve the inequality x + 3 > 2(x - 3) + 2x - 6 (Answer x < 5).