Questions: Solve the inequality. -3(1-x)+x ≤ -(9-2x)+9 The solution set is

Transcript text: Solve the inequality. \[ -3(1-x)+x \leq-(9-2 x)+9 \] The solution set is $\square$

Solution

Solution Steps

To solve the inequality $-3(1-x) + x \leq -(9-2x) + 9$, we need to simplify both sides of the inequality and then isolate the variable $x$.

Distribute the constants inside the parentheses.
Combine like terms on both sides of the inequality.
Isolate $x$ on one side of the inequality.
Solve for $x$ and express the solution in interval notation.

Step 1: Simplifying the Inequality

We start with the inequality: \[ -3(1-x) + x \leq -(9-2x) + 9 \] Distributing the terms gives: \[ -3 + 3x + x \leq -9 + 2x + 9 \] This simplifies to: \[ 4x - 3 \leq 2x \]

Step 2: Isolating the Variable

Next, we isolate $x$ by subtracting $2x$ from both sides: \[ 4x - 2x - 3 \leq 0 \] This simplifies to: \[ 2x - 3 \leq 0 \] Adding $3$ to both sides results in: \[ 2x \leq 3 \] Dividing both sides by $2$ gives: \[ x \leq \frac{3}{2} \]

Step 3: Expressing the Solution in Interval Notation

The solution set for the inequality is all values of $x$ that satisfy $x \leq \frac{3}{2}$. In interval notation, this is expressed as: \[ (-\infty, \frac{3}{2}] \]

Final Answer

$\boxed{(-\infty, \frac{3}{2}]}$

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