Evaluate -2/3(6x-12)< 1/2(8-4x)
The question presents an inequality that requires you to simplify and solve for the variable x. Specifically, you are asked to determine the values of x that satisfy the inequality once you have distributed the constants (-2/3 and 1/2) across the terms within their respective parentheses, and then collected like terms. This will involve arithmetic operations and potentially reversing the inequality sign if you multiply or divide by a negative number. The solution will give you a range or ranges of values for x that make the initial inequality true.
$- \frac{2}{3} \left(\right. 6 x - 12 \left.\right) < \frac{1}{2} \left(\right. 8 - 4 x \left.\right)$
Answer
Solution:
Step 1: Simplify the left-hand side of the inequality
Step 1.1: Express the inequality with zero addition
$0 - \frac{2}{3} \cdot (6x - 12) < \frac{1}{2} \cdot (8 - 4x)$
Step 1.2: Maintain the inequality with the addition of zeros
$- \frac{2}{3} \cdot (6x - 12) < \frac{1}{2} \cdot (8 - 4x)$
Step 1.3: Distribute $-\frac{2}{3}$ across the parentheses
$- \frac{2}{3} \cdot 6x + \frac{2}{3} \cdot 12 < \frac{1}{2} \cdot (8 - 4x)$
Step 1.4: Simplify by reducing common factors
Step 1.4.1: Adjust the negative sign in the numerator
$-\frac{2}{3} \cdot 6x + \frac{2}{3} \cdot 12 < \frac{1}{2} \cdot (8 - 4x)$
Step 1.4.2: Extract the factor of 3 from $6x$
$-\frac{2}{3} \cdot 3 \cdot 2x + \frac{2}{3} \cdot 12 < \frac{1}{2} \cdot (8 - 4x)$
Step 1.4.3: Eliminate the common factor of 3
$-\frac{2}{\cancel{3}} \cdot \cancel{3} \cdot 2x + \frac{2}{3} \cdot 12 < \frac{1}{2} \cdot (8 - 4x)$
Step 1.4.4: Rewrite the simplified expression
$-2 \cdot 2x + \frac{2}{3} \cdot 12 < \frac{1}{2} \cdot (8 - 4x)$
Step 1.5: Multiply $-2$ by $2x$
$-4x + \frac{2}{3} \cdot 12 < \frac{1}{2} \cdot (8 - 4x)$
Step 1.6: Simplify by reducing common factors
Step 1.6.1: Adjust the negative sign in the numerator
$-4x - \frac{2}{3} \cdot 12 < \frac{1}{2} \cdot (8 - 4x)$
Step 1.6.2: Extract the factor of 3 from $-12$
$-4x - \frac{2}{3} \cdot 3 \cdot (-4) < \frac{1}{2} \cdot (8 - 4x)$
Step 1.6.3: Eliminate the common factor of 3
$-4x - \frac{2}{\cancel{3}} \cdot \cancel{3} \cdot (-4) < \frac{1}{2} \cdot (8 - 4x)$
Step 1.6.4: Rewrite the simplified expression
$-4x + 2 \cdot (-4) < \frac{1}{2} \cdot (8 - 4x)$
Step 1.7: Multiply $2$ by $-4$
$-4x - 8 < \frac{1}{2} \cdot (8 - 4x)$
Step 2: Simplify the right-hand side of the inequality
Step 2.1: Distribute $\frac{1}{2}$ across the parentheses
$-4x - 8 < \frac{1}{2} \cdot 8 - \frac{1}{2} \cdot 4x$
Step 2.2: Simplify by reducing common factors
Step 2.2.1: Extract the factor of 2 from $8$
$-4x - 8 < \frac{1}{2} \cdot 2 \cdot 4 - \frac{1}{2} \cdot 4x$
Step 2.2.2: Eliminate the common factor of 2
$-4x - 8 < \frac{1}{\cancel{2}} \cdot \cancel{2} \cdot 4 - \frac{1}{2} \cdot 4x$
Step 2.2.3: Rewrite the simplified expression
$-4x - 8 < 4 - \frac{1}{2} \cdot 4x$
Step 2.3: Simplify by reducing common factors
Step 2.3.1: Extract the factor of 2 from $-4x$
$-4x - 8 < 4 - \frac{1}{2} \cdot 2 \cdot (-2x)$
Step 2.3.2: Eliminate the common factor of 2
$-4x - 8 < 4 - \frac{1}{\cancel{2}} \cdot \cancel{2} \cdot (-2x)$
Step 2.3.3: Rewrite the simplified expression
$-4x - 8 < 4 + 2x$
Step 3: Isolate terms with $x$ on one side
Step 3.1: Add $2x$ to both sides
$-4x + 2x - 8 < 4$
Step 3.2: Combine like terms
$-2x - 8 < 4$
Step 4: Isolate constant terms on the other side
Step 4.1: Add $8$ to both sides
$-2x < 4 + 8$
Step 4.2: Combine constants
$-2x < 12$
Step 5: Solve for $x$
Step 5.1: Divide by $-2$ and reverse the inequality sign
$x > \frac{12}{-2}$
Step 5.2: Simplify the left side
$x > -6$
Step 5.3: Simplify the right side
$x > -6$
Step 6: Present the solution in different forms
Inequality Form: $x > -6$ Interval Notation: $(-6, \infty)$
Knowledge Notes:
To solve the inequality $- \frac{2}{3} (6x - 12) < \frac{1}{2} (8 - 4x)$, we follow these steps:
Simplify both sides of the inequality by distributing the coefficients and combining like terms.
Isolate the variable terms on one side and the constant terms on the other side.
Solve for the variable by performing operations that maintain the inequality's balance. When dividing or multiplying by a negative number, the inequality sign must be flipped.
Express the solution in inequality form and interval notation.
Relevant algebraic concepts include:
The distributive property: $a(b + c) = ab + ac$
Combining like terms: terms with the same variable and exponent can be combined.
Inequalities: when multiplying or dividing by a negative number, the direction of the inequality sign is reversed.
Interval notation: represents the set of numbers between two endpoints. For example, $(a, b)$ represents all numbers greater than $a$ and less than $b$.
