Math  /  Algebra

QuestionIs (2,7)(2,7) a solution to this system of inequalities? y>12x+1y>5x5\begin{array}{l} y>\frac{1}{2} x+1 \\ y>5 x-5 \end{array} yes no

Studdy Solution

STEP 1

What is this asking? Does the point (2,7) (2,7) make both inequalities true? Watch out! Make sure you test *both* inequalities, not just one!

STEP 2

1. Test the first inequality.
2. Test the second inequality.

STEP 3

We're given the point (2,7) (2,7) , which means x=2 x = 2 and y=7 y = 7 .
Let's **plug** these values into the **first inequality**: y>12x+1 y > \frac{1}{2} x + 1 becomes 7>122+1 7 > \frac{1}{2} \cdot 2 + 1

STEP 4

Let's **simplify** the right side: 7>122+1 7 > \frac{1}{2} \cdot 2 + 1 7>1+1 7 > 1 + 1 7>2 7 > 2

STEP 5

Is 7>2 7 > 2 ?
Yes! So the **first inequality holds true**!
Onto the next one!

STEP 6

Now, let's **test** the **second inequality** with x=2 x = 2 and y=7 y = 7 : y>5x5 y > 5x - 5 becomes 7>525 7 > 5 \cdot 2 - 5

STEP 7

Let's **simplify** the right side: 7>525 7 > 5 \cdot 2 - 5 7>105 7 > 10 - 5 7>5 7 > 5

STEP 8

Is 7>5 7 > 5 ?
Yes! The **second inequality is also true**!

STEP 9

Since *both* inequalities are true for the point (2,7) (2,7) , the answer is **yes**!

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