Solve the Inequality for g 6(-6g+6)> -2(19g-17)

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: $(-1,\mathrm{\infty })$

Knowledge Notes:

To solve a linear inequality similar to $6(-6g+6)>-2(19g-17)$, we follow these steps:

1. Simplification: Simplify each side of the inequality separately by applying the distributive property and combining like terms.

2. Isolation: Move all terms containing the variable to one side and constants to the other side to isolate the variable.

3. Division: Divide by the coefficient of the variable to solve for the variable.

4. Solution Representation: The solution can be represented in inequality form, interval notation, or graphically on a number line.

5. Distributive Property: This property states that $a(b+c)=ab+ac$ and is used to simplify expressions.

6. Combining Like Terms: Terms with the same variable can be combined by adding or subtracting their coefficients.

7. 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.

8. Interval Notation: This is a way of writing the set of all numbers between two endpoints. For example, $(-1,\mathrm{\infty })$ means all numbers greater than $-1$.