Question

solve this system of inequalities graphically and determine one possible
inequality 1: ( y geq -x + 16 )
inequality 2: ( y leq -2.5 + 30 )
(graph with shaded regions)
answer attempt 2 out of 2
abdoulaye could have 4 quarters and 12 dimes.

Explanation:

Step1: Analyze Inequality 1

Inequality 1 is \( y \geq -x + 16 \). The line \( y=-x + 16 \) has a slope of \(-1\) and a y - intercept of \(16\). The region \( y\geq -x + 16 \) is above this line (including the line itself). From the graph, the purple - shaded region (part of it) is related to this inequality.

Step2: Analyze Inequality 2

First, simplify Inequality 2: \( y\leq - 2.5x+30\) (assuming there was a typo and it should be \(y\leq - 2.5x + 30\) instead of \(y\leq - 2.5 + 30\)). The line \(y=-2.5x + 30\) has a slope of \(-2.5\) and a y - intercept of \(30\). The region \(y\leq - 2.5x + 30\) is below this line (including the line itself). The purple - shaded region in the graph is the intersection of the regions defined by the two inequalities.

Step3: Check the solution point

We are given a possible solution: \(x = 4\) (quarters) and \(y = 12\) (dimes). Let's check it in both inequalities.

• For Inequality 1: Substitute \(x = 4\) and \(y = 12\) into \(y\geq -x + 16\). We get \(12\geq - 4+16\), which simplifies to \(12\geq12\). This is true.
• For Inequality 2: Substitute \(x = 4\) and \(y = 12\) into \(y\leq - 2.5x+30\). We get \(12\leq - 2.5\times4 + 30\). Calculate the right - hand side: \(-2.5\times4+30=-10 + 30 = 20\). And \(12\leq20\), which is also true.

Answer:

Abdoulaye could have 4 quarters and 12 dimes (since the point \((4,12)\) satisfies both \(y\geq -x + 16\) and \(y\leq - 2.5x + 30\) (after correcting the second inequality) as shown by the substitution in the steps above).