What is the complete set of solutions to the inequality -\frac{2}{3}(6x - 9) + 4 > 12? — SAT

What is the complete set of solutions to the inequality $-\frac{2}{3}(6x - 9) + 4 > 12$ ?

A
$x > -\frac{1}{2}$
$x < -\frac{1}{2}$ Answer
C
$x < -\frac{7}{2}$
D
$x < -2$

Answer

The inequality is satisfied for all values of

x

such that

x < -\frac{1}{2}

Distributing

-\frac{2}{3}

across

(6x - 9)

yields

-4x + 6

. Adding 4 gives

-4x + 10 > 12

. Subtracting 10 from both sides results in

-4x > 2

. Dividing by

-4

and reversing the inequality sign yields

x < -\frac{1}{2}

Step-by-Step Solution

Distribute

-\frac{2}{3}

to both terms inside the parentheses:

(6x - 9)

-4x + 6 + 4 > 12

, which simplifies to

-4x + 10 > 12

Applying the distributive property removes the parentheses.

Subtract 10 from both sides of the inequality to isolate the term with

x

-4x > 2

Subtracting 10 from both sides maintains the inequality while simplifying the constant terms.

Divide both sides by

-4

and reverse the inequality sign.

x < -\frac{1}{2}

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality symbol.

Key Concept

Solving linear inequalities in one variable requires distributing coefficient terms, combining constants, and reversing the inequality sign when multiplying or dividing by a negative number.

Estimated Time:1m 15s

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