WEEK FOUR
SOLUTIONS OF INEQUALITIES OF TWO VARIABLES AND THE RANGE OF VALUES OF COMBINED INEQUALITIES
A linear inequality in two variables x and y is of the form: ax + by c: ax + by < c: ax + by > c ax + by c where a, b and c are constants. A solution to an inequality is any pair of number x and y that satisfies the inequality.
Example 2
Determine the solution set of 5x + 2y 17
Solution
One solution to 5x + 2y < 17 is x =2 and y = 3 because 5(2) + 2(3) = 16, which is indeed less than 17. But the pair x = 2 and y = 3 is not the only solution. As a matter of fact, there are infinitely many solutions. If the pairs of numbers x and y is a solution, then think of this pair as a point in the plane, so the set of all solutions can be thought of as a REGION in the x –y plane.
Hence, to illustrate how to determine this region, first express y in terms of x in the inequality.
3x + 2y 17
2y -5x + 17
Y +
When x = 0, y = 8.5; when y = 0, x = 3 (show in a graph)
The shaded Region is the solution set.
RANGE OF VALUES OF COMBINED INEQUALITIES
In the above diagram, x can possess any value between -4 and +3 inclusive Hence x -2 and x + 3 or -4 x and x +3
These two inequalities can be combined as a single inequality. Thus, -4 x + 3
Example 3
What is the range of values of x for which 2x + 6 > 2 and x – 4 < 1 are both satisfied?
Solution
2x + 6 > 2 x – 4 < 1
2x > 2 – 6 x < 1 + 4
2x > – 6 x < 5
X > -2
Hence x > – 2 and x < 5 or -2 < x and x < 5 or -2 < x < 5. Both inequalities are satisfied if -2 < x and x < 5 or -2 < x < 5
a) b) c)
WRAP UP AND ASSESSMETNS
AN Inequality is any statement involving one of symbols <, >, and. Simple linear inequalities can be represented on a number line.
Ticket out ;Pg 207 Exercise 15.1 no 2d, f & No 8