Solve the Inequality for g 6(-6g+6)> -2(19g-17)
The question presents an inequality to solve for the variable \( g \). The inequality is given by the expression \( 6(-6g+6)>-2(19g-17) \), and the task is to perform the necessary mathematical operations, such as distributing, combining like terms, and isolating the variable to determine the values of \( g \) that satisfy the given inequality.
$6 \left(\right. - 6 g + 6 \left.\right) > - 2 \left(\right. 19 g - 17 \left.\right)$
Answer
Solution:
Step 1: Simplify the left-hand side of the inequality $6(-6g + 6)$.
Step 1.1: Introduce zero addition to the inequality.
$0 + 0 + 6(-6g + 6) > -2(19g - 17)$
Step 1.2: Maintain the inequality with the added zeros.
$6(-6g + 6) > -2(19g - 17)$
Step 1.3: Distribute $6$ across the terms inside the parentheses.
$6(-6g) + 6 \cdot 6 > -2(19g - 17)$
Step 1.4: Perform the multiplication.
Step 1.4.1: Multiply $-6$ by $6$.
$-36g + 6 \cdot 6 > -2(19g - 17)$
Step 1.4.2: Multiply $6$ by $6$.
$-36g + 36 > -2(19g - 17)$
Step 2: Simplify the right-hand side of the inequality $-2(19g - 17)$.
Step 2.1: Apply the distributive property.
$-36g + 36 > -2 \cdot 19g - 2 \cdot (-17)$
Step 2.2: Carry out the multiplication.
Step 2.2.1: Multiply $19$ by $-2$.
$-36g + 36 > -38g - 2 \cdot (-17)$
Step 2.2.2: Multiply $-2$ by $-17$.
$-36g + 36 > -38g + 34$
Step 3: Isolate terms with $g$ on one side.
Step 3.1: Add $38g$ to both sides.
$-36g + 36 + 38g > 34$
Step 3.2: Combine like terms.
$2g + 36 > 34$
Step 4: Move constants to the other side.
Step 4.1: Subtract $36$ from both sides.
$2g > 34 - 36$
Step 4.2: Calculate the difference.
$2g > -2$
Step 5: Solve for $g$ by dividing by the coefficient of $g$.
Step 5.1: Divide the inequality by $2$.
$\frac{2g}{2} > \frac{-2}{2}$
Step 5.2: Simplify the left-hand side.
Step 5.2.1: Reduce the fraction by canceling common factors.
Step 5.2.1.1: Cancel the $2$s.
$\frac{\cancel{2}g}{\cancel{2}} > \frac{-2}{2}$
Step 5.2.1.2: Simplify to $g$.
$g > \frac{-2}{2}$
Step 5.3: Simplify the right-hand side.
Step 5.3.1: Divide $-2$ by $2$.
$g > -1$
Step 6: Express the solution in different forms.
Inequality Form: $g > -1$ Interval Notation: $(-1, \infty)$
Knowledge Notes:
To solve a linear inequality similar to $6(-6g + 6) > -2(19g - 17)$, we follow these steps:
Simplification: Simplify each side of the inequality separately by applying the distributive property and combining like terms.
Isolation: Move all terms containing the variable to one side and constants to the other side to isolate the variable.
Division: Divide by the coefficient of the variable to solve for the variable.
Solution Representation: The solution can be represented in inequality form, interval notation, or graphically on a number line.
Distributive Property: This property states that $a(b + c) = ab + ac$ and is used to simplify expressions.
Combining Like Terms: Terms with the same variable can be combined by adding or subtracting their coefficients.
Inequalities: When dividing or multiplying both sides of an inequality by a negative number, the inequality sign must be flipped. This does not apply to our problem since we divide by a positive number.
Interval Notation: This is a way of writing the set of all numbers between two endpoints. For example, $(-1, \infty)$ means all numbers greater than $-1$.
