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Math  /  Algebra

Question1. Consider the inequality q+58>q3+2\frac{q+5}{8}>\frac{q}{3}+2. Which value of qq is a solution to the inequality? A. q=6q=6 B. q=3q=3 C. q=−11q=-11 D. q=−6q=-6

Studdy Solution

STEP 1

1. We are dealing with a linear inequality.
2. We will solve the inequality by isolating the variable q q .
3. We will check each option to see which satisfies the inequality.

STEP 2

1. Clear the fractions by finding a common denominator.
2. Simplify the inequality.
3. Solve for q q .
4. Verify which option satisfies the inequality.

STEP 3

First, we need to eliminate the fractions. The denominators are 8 and 3. The least common multiple of 8 and 3 is 24. Multiply every term by 24 to clear the fractions:
24⋅q+58>24⋅q3+24⋅2 24 \cdot \frac{q+5}{8} > 24 \cdot \frac{q}{3} + 24 \cdot 2
This simplifies to:
3(q+5)>8q+48 3(q + 5) > 8q + 48

STEP 4

Distribute and simplify the inequality:
3q+15>8q+48 3q + 15 > 8q + 48

STEP 5

Isolate the variable q q by first subtracting 3q 3q from both sides:
15>5q+48 15 > 5q + 48
Next, subtract 48 from both sides:
15−48>5q 15 - 48 > 5q −33>5q -33 > 5q
Divide both sides by 5 to solve for q q :
−335>q -\frac{33}{5} > q
This implies:
q<−335 q < -\frac{33}{5}

STEP 6

Now, we need to verify which option satisfies the inequality q<−335 q < -\frac{33}{5} .
Calculate −335-\frac{33}{5}:
−335=−6.6 -\frac{33}{5} = -6.6
Check each option: - A. q=6 q = 6 (Not less than −6.6-6.6) - B. q=3 q = 3 (Not less than −6.6-6.6) - C. q=−11 q = -11 (Less than −6.6-6.6) - D. q=−6 q = -6 (Not less than −6.6-6.6)
The value of q q that satisfies the inequality is:
−11 \boxed{-11}

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