Sure, let's solve the given problem step-by-step.
### 1. Form a System of Inequalities
Given the conditions in the problem:
1. The sum of two numbers (let's denote them as \(\( x \)\) and \(\( y \)\)) must be 24 or greater:
\(\[ x + y \geq 24 \]\)
2. The product of the two numbers must be less than 60:
\(\[ xy < 60 \]\)
So, our system of inequalities is:
\(\[ \begin{cases}
x + y \geq 24 \\
xy < 60
\end{cases} \]\)
### 2. Graph the System of Inequalities
To graph these inequalities, we'll follow these steps:
#### Step 1: Graph the Inequality \(\( x + y \geq 24 \)\)
This is a linear inequality. First, graph the boundary line \(\( x + y = 24 \)\).
- Find the x-intercept: let \(\( y = 0 \)\)
\(\[ x + 0 = 24 \Rightarrow x = 24 \]\)
- Find the y-intercept: let \(\( x = 0 \)\)
\(\[ 0 + y = 24 \Rightarrow y = 24 \]\)
Now, plot the points (24, 0) and (0, 24) on the graph. Draw a straight line passing through these points.
- Since the inequality is \(\( x + y \geq 24 \)\), shade the region above the line, including the line itself as it is 'greater than or equal to'.
#### Step 2: Graph the Inequality \(\( xy < 60 \)\)
This is a non-linear inequality. First, graph the boundary line \(\( xy = 60 \)\).
To get a sense of the curve:
- When \(\( x = 1 \)\), \(\( y = 60 \)\)
- When \(\( x = 2 \)\), \(\( y = 30 \)\)
- When \(\( x = 3 \)\), \(\( y \approx 20 \)\)
- Continue plotting points to sketch the curve of the hyperbola.
Now, draw the curve \(\( xy = 60 \)\).
- Since the inequality is \(\( xy < 60 \)\), shade the region below the curve.
#### Step 3: Intersection of the Two Inequalities
To find the solution to the system of inequalities, we identify the region that satisfies both inequalities simultaneously:
1. The region above and including the line \(\( x + y = 24 \)\).
2. The region below the hyperbola \(\( xy = 60 \)\).
### Graphical Representation
On a Cartesian plane:
- Line \(\( x + y = 24 \)\): Draw a straight line through points (24, 0) and (0, 24). Shade the half-plane above this line.
- Curve \(\( xy = 60 \)\): Draw a hyperbolic curve passing through points like (1, 60), (2, 30), etc. Shade the region under this curve.
The feasible solution area is where these two shaded regions overlap.
### Summary
By shading the regions mentioned above and looking at their intersection, you can find all the pairs of numbers (x, y) that satisfy both conditions:
\(\[ \begin{cases}
x + y \geq 24 \\
xy < 60
\end{cases} \]\)
You should see that this region is where the sum of \(\( x \)\) and \(\( y \)\) is at least 24 and their product is less than 60. The intersection will form a feasible region meeting both inequalities.
