if X is an integer, write down the largest possible value of X that satisfies the inequality
Answers
Answer:
A graphical approach
First, we sketch the three lines
2x+3yx+23y+1=23,=3yand=4x,2x+3y=23,x+2=3yand3y+1=4x,
by rearranging them in the form y=mx+cy=mx+c first.
Then thinking about which side of each of the lines is satisfied by the inequalities, we see that the central triangular region is the one satisfied by all three inequalities.
We can see from the graph that the largest possible value of xx is where the lines 2x+3y=232x+3y=23 and x+2=3yx+2=3y intersect, which is at the point (7,3)(7,3).
So the answer is (b).
........................2nd one....................................
An algebraic approach
We want to find the largest possible value of xx, so we aim to eliminate yy.
First attempt:
The third inequality rearranges to give 3y≤4x−13y≤4x−1.
We cannot substitute this into the first inequality, as the directions of the inequalities are not compatible
.
If we substitute it into the second inequality, though, we obtain x+2≤4x−1x+2≤4x−1, so 3x≥33x≥3, or x≥1x≥1, which is not so helpful.
Second attempt:
The first inequality rearranges to give 3y≤23−2x3y≤23−2x. Combining this with the second inequality gives
x+2≤3y≤23−2xx+2≤3y≤23−2x
so 3x≤213x≤21 or x≤7x≤7.
If x=7x=7, the second inequality gives 9≤3y9≤3y, so we could try y=3y=3. This pair of values satisfies all three inequalities, so the greatest possible value of xx is 77, and the answer is (b).
