8x−8≤−72.
The Correct Answer and Explanation is:
To solve the inequality ( 8x – 8 \leq -72 ), we can follow these steps:
Step 1: Isolate the term with ( x )
Start by moving the constant term on the left side to the right side. You do this by adding 8 to both sides of the inequality:
[
8x – 8 + 8 \leq -72 + 8
]
This simplifies to:
[
8x \leq -64
]
Step 2: Solve for ( x )
Now, to isolate ( x ), divide both sides of the inequality by 8:
[
\frac{8x}{8} \leq \frac{-64}{8}
]
This simplifies to:
[
x \leq -8
]
Step 3: Write the final solution
The solution to the inequality is ( x \leq -8 ). This means that ( x ) can be any value less than or equal to -8.
Explanation
When solving an inequality like ( 8x – 8 \leq -72 ), the process is similar to solving an equation, but you need to be careful when multiplying or dividing by negative numbers, as this will flip the direction of the inequality. In this case, no negative number was involved in the division step, so the inequality direction remains the same.
- The first step involves moving constants to one side to isolate the variable term, which in this case was achieved by adding 8 to both sides.
- After simplifying, we divided both sides by 8 to solve for ( x ), ensuring that we followed the order of operations (PEMDAS/BODMAS) and treated both sides equally.
- The final solution, ( x \leq -8 ), is the set of all real numbers that satisfy the inequality.
Graphical Interpretation
On a number line, this solution is represented by a shaded region that includes -8 and extends to the left, indicating that any number less than or equal to -8 satisfies the inequality.