solve 3x<2y+5,8y-9x>10

Answer 1

To solve the system of inequalities given by $3x<2y+5$ and $8y-9x>10$, we will rewrite each inequality into a clearer form and then identify the region that satisfies both.

Step 1: Rearrange the inequalities

1. Rearranging the first inequality: $$3x<2y+5\Rightarrow 2y>3x-5\Rightarrow y>\frac{3}{2}x-\frac{5}{2}$$

2. Rearranging the second inequality: $$8y-9x>10\Rightarrow 8y>9x+10\Rightarrow y>\frac{9}{8}x+\frac{10}{8}\Rightarrow y>\frac{9}{8}x+\frac{5}{4}$$

Step 2: Graph the inequalities

1. Graphing $y>\frac{3}{2}x-\frac{5}{2}$:

   • The line $y=\frac{3}{2}x-\frac{5}{2}$ has a y-intercept at $-\frac{5}{2}$ and a slope of $\frac{3}{2}$.
   • The region above this line represents where $y>\frac{3}{2}x-\frac{5}{2}$.

2. Graphing $y>\frac{9}{8}x+\frac{5}{4}$:

   • The line $y=\frac{9}{8}x+\frac{5}{4}$ has a y-intercept at $\frac{5}{4}$ and a slope of $\frac{9}{8}$.
   • The region above this line represents where $y>\frac{9}{8}x+\frac{5}{4}$.

Step 3: Identify the solution region

The solution to the system of inequalities will be the region that lies above both lines. To visualize this:

• Plot both lines on a graph.
• Shade the areas above each line.

Step 4: Summary of the solution

The solution set consists of all the points $(x,y)$ that lie in the region where both conditions are satisfied (above both lines).

If you're looking to find specific points or coordinates that satisfy both inequalities, you can test various points within the shaded area.

For example:

• Choose $x=0$:
  • For the first inequality: $y>-\frac{5}{2}$ → $y>-2.5$
  • For the second inequality: $y>\frac{5}{4}$ → $y>1.25$

A point like $(0,2)$ satisfies both inequalities as it is above both lines:

$(0,2)$ works since:

• $2>-2.5$ (first inequality)
• $2>1.25$ (second inequality)

Thus, the final solution represents all points $(x,y)$ for which these conditions hold true within the specified shaded region on the graph.