Question

which of the following values are solutions to the inequality 2 ≥ -1 - 2x? i. 3 ii. -5 iii. -6 answer none i only ii only iii only i and ii i and iii ii and iii i, ii and iii submit answer

Explanation:

Step1: Solve the inequality

Start with \( 2 \geq -1 - 2x \). Add 1 to both sides: \( 2 + 1 \geq -2x \), so \( 3 \geq -2x \). Divide both sides by -2 (remember to reverse the inequality sign): \( \frac{3}{-2} \leq x \), which simplifies to \( x \geq -\frac{3}{2} = -1.5 \).

Step2: Check each value

• I. 3: \( 3 \geq -1.5 \), true.
• II. -5: \( -5 \geq -1.5 \)? No, false.
• III. -6: \( -6 \geq -1.5 \)? No, false. Wait, wait, no—wait, when we divided by -2, let's recheck the solving. Wait, original inequality: \( 2 \geq -1 - 2x \). Add 1: \( 3 \geq -2x \). Divide by -2: inequality flips, so \( x \geq -\frac{3}{2} \) (since dividing by negative reverses the inequality). So \( x \geq -1.5 \). Now check:
• I. 3: 3 ≥ -1.5: yes.
• II. -5: -5 ≥ -1.5? No, -5 is less than -1.5.
• III. -6: -6 ≥ -1.5? No. Wait, that can't be. Wait, maybe I made a mistake in solving. Let's do it again. \( 2 \geq -1 - 2x \). Add 2x to both sides: \( 2 + 2x \geq -1 \). Subtract 2: \( 2x \geq -3 \). Divide by 2: \( x \geq -\frac{3}{2} = -1.5 \). Yes, that's correct. So x must be greater than or equal to -1.5. So 3 is ≥ -1.5 (yes), -5 is -5 < -1.5 (no), -6 < -1.5 (no). Wait, but that would mean only I is correct. But let's check by plugging into the original inequality.

Check I: 3. Plug into \( 2 \geq -1 - 2(3) \): \( 2 \geq -1 -6 = -7 \). 2 ≥ -7: yes, true.

Check II: -5. Plug into \( 2 \geq -1 -2(-5) = -1 +10 =9 \). 2 ≥9? No, false.

Check III: -6. Plug into \( 2 \geq -1 -2(-6) = -1 +12 =11 \). 2 ≥11? No, false. Wait, so only I is correct? But wait, maybe I messed up the solving. Wait, let's solve again:

\( 2 \geq -1 - 2x \)

Add 2x to both sides: \( 2 + 2x \geq -1 \)

Subtract 2: \( 2x \geq -3 \)

Divide by 2: \( x \geq -1.5 \). Correct. So 3 ≥ -1.5: yes. -5 ≥ -1.5: no. -6 ≥ -1.5: no. So only I is correct. Wait, but the options have "I only". Wait, but let me check again. Wait, maybe I made a mistake in the sign when moving terms. Let's try another approach. Let's solve for x:

\( 2 \geq -1 - 2x \)

Add 2x to both sides: \( 2 + 2x \geq -1 \)

Subtract 2: \( 2x \geq -3 \)

Divide by 2: \( x \geq -1.5 \). Correct. So x must be ≥ -1.5. So 3 is ≥ -1.5 (yes), -5 is -5 < -1.5 (no), -6 < -1.5 (no). So only I is correct. So the answer is I only.

Answer:

I only