Below are two inequalities and the graphs of their lines *without* the shading. By imagining where the shading should be, identify which point would satisfy BOTH inequalities.

y, is less than, minus, 2, x, plus, 2

*y*<−2*x*+2

y, is less than, start fraction, 3, divided by, 4, end fraction, x, plus, 1

*y*<43​*x*+1

12345678910-1-2-3-4-5-6-7-8-9-1012345678910-1-2-3-4-5-6-7-8-9-10xy

Answer

Multiple Choice Answers

left parenthesis, 2, comma, 5, right parenthesis(2,5)

left parenthesis, minus, 10, comma, 5, right parenthesis(−10,5)

left parenthesis, 7, comma, minus, 5, right parenthesis(7,−5)

left parenthesis, minus, 2, comma, minus, 10, right parenthesis(−2,−10)

To solve the problem, we need to identify which of the given points satisfies both inequalities. Let's break down the inequalities and test each point.

### Inequality 1:
\[ y < -2x + 2 \]

This represents a line with a slope of -2 and a y-intercept of 2. The shading would be below this line.

### Inequality 2:
\[ y < \frac{3}{4}x + 1 \]

This represents a line with a slope of \( \frac{3}{4} \) and a y-intercept of 1. The shading would also be below this line.

We need to test each of the given points to see if they satisfy **both inequalities**.

### Point (2, 5):
For inequality 1:  
\[ y < -2(2) + 2 \]
\[ 5 < -4 + 2 \]
\[ 5 < -2 \] (False)

So, point (2, 5) does **not** satisfy both inequalities.

### Point (-10, 5):
For inequality 1:  
\[ 5 < -2(-10) + 2 \]
\[ 5 < 20 + 2 \]
\[ 5 < 22 \] (True)

For inequality 2:  
\[ 5 < \frac{3}{4}(-10) + 1 \]
\[ 5 < -7.5 + 1 \]
\[ 5 < -6.5 \] (False)

So, point (-10, 5) does **not** satisfy both inequalities.

### Point (7, -5):
For inequality 1:  
\[ -5 < -2(7) + 2 \]
\[ -5 < -14 + 2 \]
\[ -5 < -12 \] (False)

So, point (7, -5) does **not** satisfy both inequalities.

### Point (-2, -10):
For inequality 1:  
\[ -10 < -2(-2) + 2 \]
\[ -10 < 4 + 2 \]
\[ -10 < 6 \] (True)

For inequality 2:  
\[ -10 < \frac{3}{4}(-2) + 1 \]
\[ -10 < -1.5 + 1 \]
\[ -10 < -0.5 \] (True)

So, point (-2, -10) **does** satisfy both inequalities.

### Conclusion:
The correct answer is **(-2, -10)**.

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Would you like more details about how these inequalities and points were evaluated, or have any questions?

Here are 5 related questions to expand this topic:
1. How do you graph inequalities on a coordinate plane?
2. What is the significance of the slope and y-intercept in a linear equation?
3. Can inequalities ever have the same solution region?
4. How do you determine which side of the line to shade in an inequality graph?
5. How do systems of inequalities differ from systems of equations?

**Tip:** When solving inequalities, always remember to test points in the region you're interested in, to confirm they satisfy all conditions.

Below are two inequalities and the graphs of their lines without the shading. By imagining where the shading should be, identify which point would satisfy BOTH inequalities.

y < -2x + 2
y < (3/4)x + 1

Multiple Choice Answers:
(2, 5)
(-10, 5)
(7, -5)
(-2, -10)

Learn how to solve systems of linear inequalities using two equations: y < -2x + 2 and y < (3/4)x + 1. This problem tests your understanding of linear inequality graphs and points that satisfy both conditions. Suitable for students in Grades 8-10.

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