which of the following points shows a solution to the simultaneous inequalities y > 1/2 x + 5 and y < -2x + 1?

Question

a (1,5.5)
b (-5,5)
c (4,8)
d (-1.6,4.2)

Answer 1

To find the points that satisfy the simultaneous inequalities $y>\frac{1}{2}x+5$ and $y<-2x+1$, we will evaluate each point one by one.

1. For point a (1, 5.5):

   • Check $y>\frac{1}{2}x+5$: $$5.5>\frac{1}{2}(1)+5\Rightarrow 5.5>0.5+5\Rightarrow 5.5>5.5\text{ (False)}$$
   • So, point a does not satisfy the inequalities.

2. For point b (-5, 5):

   • Check $y>\frac{1}{2}x+5$: $$5>\frac{1}{2}(-5)+5\Rightarrow 5>-2.5+5\Rightarrow 5>2.5\text{ (True)}$$
   • Check $y<-2x+1$: $$5<-2(-5)+1\Rightarrow 5<10+1\Rightarrow 5<11\text{ (True)}$$
   • Point b satisfies both inequalities.

3. For point c (4, 8):

   • Check $y>\frac{1}{2}x+5$: $$8>\frac{1}{2}(4)+5\Rightarrow 8>2+5\Rightarrow 8>7\text{ (True)}$$
   • Check $y<-2x+1$: $$8<-2(4)+1\Rightarrow 8<-8+1\Rightarrow 8<-7\text{ (False)}$$
   • Point c does not satisfy both inequalities.

4. For point d (-1.6, 4.2):

   • Check $y>\frac{1}{2}x+5$: $$4.2>\frac{1}{2}(-1.6)+5\Rightarrow 4.2>-0.8+5\Rightarrow 4.2>4.2\text{ (False)}$$
   • So, point d does not satisfy the inequalities.

In conclusion, the only point that shows a solution to the simultaneous inequalities is: b (-5, 5).