Question
Is 5x -10y < 170 a linear inequality in two variables? Tell how to solve
Answers
GPT-4o mini
Yes, the inequality \(5x - 10y < 170\) is a linear inequality in two variables, \(x\) and \(y\). A linear inequality in two variables can be expressed in the form \(Ax + By < C\), where \(A\), \(B\), and \(C\) are constants. In this case, \(A = 5\), \(B = -10\), and \(C = 170\).
To solve the inequality, you can follow these steps:
1. **Rewrite the inequality**:
Start by rewriting the inequality in a more standard form if necessary:
\[
5x - 10y < 170
\]
2. **Isolate one variable**:
You can solve for \(y\) in terms of \(x\) by isolating \(y\):
\[
-10y < -5x + 170
\]
Now divide by -10. Remember, when you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality sign:
\[
y > \frac{1}{2}x - 17
\]
3. **Graph the line**:
The related equation \(y = \frac{1}{2}x - 17\) represents a line. To graph this line:
- Find the y-intercept by setting \(x = 0\):
\[
y = -17 \quad (0, -17)
\]
- Find the x-intercept by setting \(y = 0\):
\[
0 = \frac{1}{2}x - 17 \implies x = 34 \quad (34, 0)
\]
4. **Draw the line**:
Draw the line connecting the points \((0, -17)\) and \((34, 0)\). Since the inequality is strict (\(<\)), you will use a dashed line to indicate that points on the line itself are not included in the solution.
5. **Determine the shading**:
You need to determine which side of the line to shade. Choose a test point that is not on the line, such as \((0, 0)\):
\[
5(0) - 10(0) < 170 \implies 0 < 170 \quad \text{(True)}
\]
Since the test point \((0, 0)\) satisfies the inequality, shade the region that includes \((0, 0)\). This region represents all the solutions satisfying \(5x - 10y < 170\).
In conclusion, the solution to the inequality \(5x - 10y < 170\) is the region above the line \(y = \frac{1}{2}x - 17\) in the coordinate plane, excluding the line itself.
To solve the inequality, you can follow these steps:
1. **Rewrite the inequality**:
Start by rewriting the inequality in a more standard form if necessary:
\[
5x - 10y < 170
\]
2. **Isolate one variable**:
You can solve for \(y\) in terms of \(x\) by isolating \(y\):
\[
-10y < -5x + 170
\]
Now divide by -10. Remember, when you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality sign:
\[
y > \frac{1}{2}x - 17
\]
3. **Graph the line**:
The related equation \(y = \frac{1}{2}x - 17\) represents a line. To graph this line:
- Find the y-intercept by setting \(x = 0\):
\[
y = -17 \quad (0, -17)
\]
- Find the x-intercept by setting \(y = 0\):
\[
0 = \frac{1}{2}x - 17 \implies x = 34 \quad (34, 0)
\]
4. **Draw the line**:
Draw the line connecting the points \((0, -17)\) and \((34, 0)\). Since the inequality is strict (\(<\)), you will use a dashed line to indicate that points on the line itself are not included in the solution.
5. **Determine the shading**:
You need to determine which side of the line to shade. Choose a test point that is not on the line, such as \((0, 0)\):
\[
5(0) - 10(0) < 170 \implies 0 < 170 \quad \text{(True)}
\]
Since the test point \((0, 0)\) satisfies the inequality, shade the region that includes \((0, 0)\). This region represents all the solutions satisfying \(5x - 10y < 170\).
In conclusion, the solution to the inequality \(5x - 10y < 170\) is the region above the line \(y = \frac{1}{2}x - 17\) in the coordinate plane, excluding the line itself.