Solve The Inequality: X - 9 \leq 2(9 - X)

When you're faced with an inequality like $x-9 \leq 2(9-x)$, the goal is to find the range of values for the variable '$x$' that make the statement true. Think of it like a balancing act; whatever you do to one side of the inequality, you must do to the other to keep it balanced. We'll walk through the steps to isolate '$x$' and determine its possible values.

Understanding Inequalities

Before we dive into solving this specific inequality, let's quickly touch upon what inequalities are and how they work. Unlike equations that have a single solution (or sometimes no solution), inequalities represent a range of solutions. The symbols we use are:

• < (less than)
• > (greater than)
• ≤ (less than or equal to)
• ≥ (greater than or equal to)

The fundamental rule when working with inequalities is similar to equations: you can add, subtract, multiply, or divide both sides by the same number. However, there's a crucial exception: if you multiply or divide both sides by a negative number, you must flip the direction of the inequality sign. This is a common pitfall, so always keep it in mind!

Step-by-Step Solution

Let's break down the inequality $x-9 \leq 2(9-x)$ piece by piece:

Step 1: Distribute on the right side.

Our first move is to simplify the right side of the inequality by distributing the '2' to both terms inside the parentheses:

$x - 9 \leq 2 \times 9 - 2 \times x$

$x - 9 \leq 18 - 2x$

Step 2: Gather '$x$' terms on one side.

Now, we want to get all the terms containing '$x$' onto one side of the inequality. It's often easiest to move them to the side where the '$x$' term will have a positive coefficient. In this case, adding $2x$ to both sides will achieve that:

$x - 9 + 2x \leq 18 - 2x + 2x$

Combine the '$x$' terms on the left:

$3x - 9 \leq 18$

Step 3: Gather constant terms on the other side.

Next, we'll move the constant terms (the numbers without variables) to the opposite side. To do this, we'll add '9' to both sides of the inequality:

$3x - 9 + 9 \leq 18 + 9$

$3x \leq 27$

Step 4: Isolate '$x$'.

Finally, to get '$x$' by itself, we need to divide both sides by the coefficient of '$x$', which is '3'. Since '3' is a positive number, we do not need to flip the inequality sign:

$\frac{3x}{3} \leq \frac{27}{3}$

$x \leq 9$

Interpreting the Solution

So, we've found that the inequality $x \leq 9$ holds true. This means that any value of '$x$' that is less than or equal to 9 will satisfy the original inequality $x-9 \leq 2(9-x)$.

For instance, let's test a value:

• If $x = 9$: $9 - 9 \leq 2(9 - 9) \implies 0 \leq 2(0) \implies 0 \leq 0$. This is true.
• If $x = 8$: $8 - 9 \leq 2(9 - 8) \implies -1 \leq 2(1) \implies -1 \leq 2$. This is also true.
• If $x = 10$: $10 - 9 \leq 2(9 - 10) \implies 1 \leq 2(-1) \implies 1 \leq -2$. This is false.

This confirms our solution. The set of values for '$x$' that makes the inequality true is all numbers less than or equal to 9.

Conclusion

By following these straightforward algebraic steps, we've successfully solved the inequality $x-9 \leq 2(9-x)$. The solution is $x \leq 9$. This means that any number you pick that is 9 or smaller will make the original inequality a true statement. Remember to always be mindful of the rules for manipulating inequalities, especially when dealing with negative numbers!

For further exploration into solving inequalities and algebraic concepts, you can visit trusted resources like Khan Academy or Math is Fun.