What is the greatest integer value of x that satisfies the inequality \frac{7 - 5x}{3} - 4… — ACT

What is the greatest integer value of $x$ that satisfies the inequality $\frac{7 - 5x}{3} - 4 \ge 2(x + 6)$ ?

Answer: -4

The greatest integer value of

x

that satisfies the inequality is

-4

To find the greatest integer value of

x

that satisfies the inequality, solve the inequality algebraically. First, clear the fraction by multiplying all terms by 3:

7 - 5x - 12 \ge 6(x + 6)

. Simplify the left side to

-5x - 5

and distribute the right side to get

6x + 36

. Move variables to the left side by subtracting

6x

to get

-11x - 5 \ge 36

. Add 5 to both sides to get

-11x \ge 41

. Finally, divide by

-11

and reverse the inequality sign, yielding

x \le -\frac{41}{11}

, which is approximately

x \le -3.73

. The greatest integer less than or equal to

-3.73

-4

Multiply the entire inequality by 3 to clear the fraction

7 - 5x - 12 \ge 6(x + 6)

Multiplying by the common denominator eliminates the fraction and simplifies further algebraic steps.

Simplify the left side and distribute the right side

-5x - 5 \ge 6x + 36

Combines constant terms on the left side and expands the parentheses on the right side.

Subtract

6x

from both sides

-11x - 5 \ge 36

Gathers all variable terms on the left side of the inequality.

Add 5 to both sides

-11x \ge 41

Isolates the variable term by moving the constant to the right side.

Divide by -11 and reverse the inequality sign

x \le -\frac{41}{11}

Isolates the variable

x

. Reversing the inequality sign is required when multiplying or dividing both sides by a negative number.

Determine the greatest integer satisfying the inequality

x \le -3.73

, so the greatest integer is

-4

Since

-\frac{41}{11} \approx -3.73

, the largest integer that is less than or equal to this value is

-4

Solving linear inequalities, applying the sign-flip rule when dividing by a negative number, and identifying integer boundaries.

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