\begin{tabular}{l} 16. $$ y

Question

\begin{tabular}{l} 16. $$ y<2 x+1 $$ \ $$ y \leq-x-4 $$ \ \hline \ \hline\end{tabular}

UpStudy AI · Accepted Answer

To analyze the system of inequalities given:

1. $$ y < 2x + 1 $$
2. $$ y \leq -x - 4 $$

We can graph these inequalities to find the solution region.

### Step 1: Graph the first inequality $$ y < 2x + 1 $$

- The line $$ y = 2x + 1 $$ is a straight line with a slope of 2 and a y-intercept of 1.
- Since the inequality is strict ($$<$$), we will use a dashed line to indicate that points on the line are not included in the solution.
- The region below this line represents the solution to the inequality.

### Step 2: Graph the second inequality $$ y \leq -x - 4 $$

- The line $$ y = -x - 4 $$ has a slope of -1 and a y-intercept of -4.
- Since the inequality is non-strict ($$\leq$$), we will use a solid line to indicate that points on the line are included in the solution.
- The region below this line represents the solution to the inequality.

### Step 3: Find the intersection of the two inequalities

To find the intersection of the two lines, we can set them equal to each other:

$$
2x + 1 = -x - 4
$$

Solving for $$ x $$:

$$
2x + x = -4 - 1
$$
$$
3x = -5
$$
$$
x = -\frac{5}{3}
$$

Now, substitute $$ x = -\frac{5}{3} $$ back into one of the equations to find $$ y $$:

Using $$ y = 2x + 1 $$:

$$
y = 2\left(-\frac{5}{3}\right) + 1 = -\frac{10}{3} + 1 = -\frac{10}{3} + \frac{3}{3} = -\frac{7}{3}
$$

So the intersection point is:

$$
\left(-\frac{5}{3}, -\frac{7}{3}\right)
$$

### Step 4: Determine the solution region

The solution region is where the shaded areas of both inequalities overlap.

- The area below the dashed line $$ y = 2x + 1 $$ and below the solid line $$ y = -x - 4 $$ is the solution to the system of inequalities.

### Conclusion

The solution to the system of inequalities $$ y < 2x + 1 $$ and $$ y \leq -x - 4 $$ is the region in the coordinate plane that is below both lines, including the line $$ y = -x - 4 $$ but not including the line $$ y = 2x + 1 $$. The intersection point $$ \left(-\frac{5}{3}, -\frac{7}{3}\right) $$ is also part of the boundary of the solution region.
The solution to the system of inequalities $$ y < 2x + 1 $$ and $$ y \leq -x - 4 $$ is the region in the coordinate plane below both lines, including the line $$ y = -x - 4 $$ but not including the line $$ y = 2x + 1 $$. The intersection point is $$ \left(-\frac{5}{3}, -\frac{7}{3}\right) $$.

\begin{tabular}{l} 16. \( y<2 x+1 \) \\ \( y \leq-x-4 \) \\ \hline \\ \hline\end{tabular}

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Algebra Questions