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The points [tex]\((3, -2)\)[/tex] and [tex]\((2, 1)\)[/tex] are both solutions to which inequality?

A. [tex]\(y \geq 4x - 2\)[/tex]

B. [tex]\(y \ \textless \ -2x + 1\)[/tex]

C. [tex]\(y \ \textgreater \ \frac{1}{2}x + 2\)[/tex]

D. [tex]\(y \leq \frac{3}{2}x - 1\)[/tex]

Sagot :

GinnyAnsweridn
Sure, let's determine which inequality fits both given points [tex]\((3, -2)\)[/tex] and [tex]\((2, 1)\)[/tex].

### Step-by-Step Solution:

#### Point (3, -2):

1. Inequality [tex]\( y \geq 4x - 2 \)[/tex]:
[tex]\( y = -2 \)[/tex] and [tex]\( 4x - 2 = 4(3) - 2 = 12 - 2 = 10 \)[/tex]
- [tex]\(-2 \geq 10\)[/tex] is false.

2. Inequality [tex]\( y < -2x + 1 \)[/tex]:
[tex]\( y = -2 \)[/tex] and [tex]\( -2x + 1 = -2(3) + 1 = -6 + 1 = -5 \)[/tex]
- [tex]\(-2 < -5\)[/tex] is false.

3. Inequality [tex]\( y > \frac{1}{2}x + 2 \)[/tex]:
[tex]\( y = -2 \)[/tex] and [tex]\( \frac{1}{2}x + 2 = \frac{1}{2}(3) + 2 = 1.5 + 2 = 3.5 \)[/tex]
- [tex]\(-2 > 3.5\)[/tex] is false.

4. Inequality [tex]\( y \leq \frac{3}{2}x - 1 \)[/tex]:
[tex]\( y = -2 \)[/tex] and [tex]\( \frac{3}{2}x - 1 = \frac{3}{2}(3) - 1 = 4.5 - 1 = 3.5 \)[/tex]
- [tex]\(-2 \leq 3.5\)[/tex] is true.

#### Point (2, 1):

1. Inequality [tex]\( y \geq 4x - 2 \)[/tex]:
[tex]\( y = 1 \)[/tex] and [tex]\( 4x - 2 = 4(2) - 2 = 8 - 2 = 6 \)[/tex]
- [tex]\( 1 \geq 6\)[/tex] is false.

2. Inequality [tex]\( y < -2x + 1 \)[/tex]:
[tex]\( y = 1 \)[/tex] and [tex]\( -2x + 1 = -2(2) + 1 = -4 + 1 = -3 \)[/tex]
- [tex]\( 1 < -3\)[/tex] is false.

3. Inequality [tex]\( y > \frac{1}{2}x + 2 \)[/tex]:
[tex]\( y = 1 \)[/tex] and [tex]\( \frac{1}{2}x + 2 = \frac{1}{2}(2) + 2 = 1 + 2 = 3 \)[/tex]
- [tex]\( 1 > 3\)[/tex] is false.

4. Inequality [tex]\( y \leq \frac{3}{2}x - 1 \)[/tex]:
[tex]\( y = 1 \)[/tex] and [tex]\( \frac{3}{2}x - 1 = \frac{3}{2}(2) - 1 = 3 - 1 = 2 \)[/tex]
- [tex]\( 1 \leq 2\)[/tex] is true.

### Conclusion:

After checking both points against all four inequalities, we find that both points satisfy only the inequality [tex]\( y \leq \frac{3}{2}x - 1 \)[/tex]. Thus, the correct inequality to which both points are solutions is:

[tex]\[ y \leq \frac{3}{2}x - 1 \][/tex]

Which corresponds to choice 4.
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