`{{:(y gt -x-5),(ylt 2x +3):}`
The figure above shows the solution for the system of inequalities shown. Suppose (a,b) is a solution to the system. If `a = 0,` what is the greatest possible interger value of b ?

Answer

2

Explanation

If (a,b) is a solution to the system, then a is the x-coordinate of any p[oint in the region where the shading overlaps, and b is the corrspojnding y-coordinate. When `a =0 (or x =o), ` then maximum possible value for b lies on the upper boundary line, `y lt 2x +3.` It looks like the y-coordinate is 3, but to be sure, substitute `x =0` into the equation and simplify. You can replace the inequality symbol with an equal sign to make it an equatin because you are finding a point on the boundary line, not a range:
`y = 2 (0) +3`
` y =3`
The point on the boundary line is `(0,3).` The boundary line is dashed (because the inequality is strictly less than), so `(0,3)` is not a solution to the system. This means 2 is the greaterst possible integer value for b when `a =0.` Grid in 2.