If the solution to the inequality a(2 - 3x) - 4(x + 3) \ge 12, where a is a constant, is x… — SAT

If the solution to the inequality $a(2 - 3x) - 4(x + 3) \ge 12$ , where $a$ is a constant, is $x \le -1$ , what is the value of $a$ ?

Answer: 4

The value of

a

4

To solve the inequality

a(2 - 3x) - 4(x + 3) \ge 12

, we first expand it to get

2a - 3ax - 4x - 12 \ge 12

. Grouping the

x

terms gives

(-3a - 4)x \ge 24 - 2a

. Since the inequality's solution is

x \le -1

, the direction of the inequality must flip, which means the coefficient of

x

, namely

-3a - 4

, must be negative. Dividing both sides by this coefficient gives the boundary value of the inequality as

\frac{24 - 2a}{-3a - 4}

. Setting this boundary equal to

-1

yields

24 - 2a = 3a + 4

, which simplifies to

5a = 20

, or

a = 4

. Since

a = 4

makes the coefficient

-3(4) - 4 = -16

negative, the solution holds.

Expand the inequality using the distributive property.

2a - 3ax - 4x - 12 \ge 12

This allows us to separate and group the terms containing the variable

x

and the constant terms.

Group like terms and isolate the variable terms on the left-hand side.

(-3a - 4)x \ge 24 - 2a

By combining the coefficients of

x

and adding

12 - 2a

to both sides, we prepare the inequality to solve for

x

Determine the effect of dividing by the variable's coefficient.

Since the given solution is

x \le -1

, the inequality sign flipped from

\ge

\le

. Therefore, the coefficient

-3a - 4

must be negative.

Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

Set the boundary value of the solution equal to

-1

and solve for

a

a = 4

Setting the boundary of the inequality

\frac{24 - 2a}{-3a - 4}

equal to

-1

allows us to find the specific constant

a

that produces this solution set.

Solving linear inequalities in one variable involving parameters and sign flips.

Question