Question

what is the solution to $-4(8 - 3x) \geq 6x - 8$?\
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$\bigcirc$ $x \geq -\frac{4}{3}$\
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$\bigcirc$ $x \leq -\frac{4}{3}$\
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$\bigcirc$ $x \geq 4$\
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$\bigcirc$ $x \leq 4$

Explanation:

Step1: Distribute -4

First, we distribute -4 on the left side of the inequality:
$$-4(8 - 3x) = -32 + 12x$$
So the inequality becomes:
$$-32 + 12x \geq 6x - 8$$

Step2: Subtract 6x from both sides

Subtract \(6x\) from both sides to get:
$$-32 + 12x - 6x \geq 6x - 8 - 6x$$
Simplify:
$$-32 + 6x \geq -8$$

Step3: Add 32 to both sides

Add 32 to both sides:
$$-32 + 6x + 32 \geq -8 + 32$$
Simplify:
$$6x \geq 24$$

Step4: Divide by 6

Divide both sides by 6:
$$\frac{6x}{6} \geq \frac{24}{6}$$
Simplify:
$$x \geq 4$$

Wait, that's not matching? Wait, no, wait, let's check again. Wait, no, wait, I think I made a mistake. Wait, let's redo the steps.

Wait, original inequality: \(-4(8 - 3x) \geq 6x - 8\)

Step1: Distribute -4: \(-32 + 12x \geq 6x - 8\)

Step2: Subtract 6x from both sides: \(12x - 6x - 32 \geq -8\) => \(6x - 32 \geq -8\)

Step3: Add 32 to both sides: \(6x \geq -8 + 32\) => \(6x \geq 24\)

Step4: Divide by 6: \(x \geq 4\)? But wait, the options have x ≥4 as an option. Wait, but let's check with x=5:

Left side: -4(8 - 15) = -4(-7) = 28

Right side: 6*5 -8 = 30 -8 =22

28 ≥22: true.

x=3:

Left side: -4(8 -9)= -4(-1)=4

Right side: 18 -8=10

4 ≥10: false. So x≥4 is correct. Wait, but the options have x≥4 as option C. Wait, but let's check again.

Wait, maybe I made a mistake earlier. Wait, let's do it again:

Original inequality: \(-4(8 - 3x) \geq 6x - 8\)

Distribute -4: \(-32 + 12x \geq 6x - 8\)

Subtract 6x: \(12x -6x -32 \geq -8\) => \(6x -32 \geq -8\)

Add 32: \(6x \geq 24\)

Divide by 6: \(x \geq 4\). So the correct answer is x ≥4, which is option C: x ≥4.

Wait, but let's check with x=4:

Left side: -4(8 -12)= -4(-4)=16

Right side: 24 -8=16. 16 ≥16: true.

x=5: as before, 28 ≥22: true.

x=3: 4 ≥10: false. So yes, x≥4 is correct. So the answer is x ≥4, which is option C.

Wait, but the options are:

O x ≥ -4/3

O x ≤ -4/3

O x ≥4

O x ≤4

So the correct answer is x ≥4, which is the third option.

Wait, but let's check again. Wait, maybe I messed up the sign? Wait, no, let's check the steps again.

Original inequality: \(-4(8 - 3x) \geq 6x - 8\)

Distribute: \(-32 + 12x \geq 6x - 8\)

Subtract 6x: \(6x - 32 \geq -8\)

Add 32: \(6x \geq 24\)

Divide by 6: \(x \geq 4\). Yes, that's correct. So the answer is x ≥4, which is option C: x ≥4.

Answer:

C. \( x \geq 4 \)