Certainly! Let's solve the inequality step by step:
Given the inequality:
[tex]\[ 2(4x - 3) \geq -3(3x) + 5x \][/tex]
Step 1: Expand and simplify both sides of the inequality.
First, expand the left side:
[tex]\[ 2(4x - 3) \][/tex]
[tex]\[ = 2 \cdot 4x - 2 \cdot 3 \][/tex]
[tex]\[ = 8x - 6 \][/tex]
Next, expand the right side:
[tex]\[ -3(3x) + 5x \][/tex]
[tex]\[ = -3 \cdot 3x + 5x \][/tex]
[tex]\[ = -9x + 5x \][/tex]
[tex]\[ = -4x \][/tex]
Now we have the simplified inequality:
[tex]\[ 8x - 6 \geq -4x \][/tex]
Step 2: Combine like terms by isolating [tex]\(x\)[/tex] on one side of the inequality.
Add [tex]\(4x\)[/tex] to both sides:
[tex]\[ 8x - 6 + 4x \geq -4x + 4x \][/tex]
[tex]\[ 12x - 6 \geq 0 \][/tex]
Step 3: Isolate [tex]\(x\)[/tex] by moving the constant term to the other side.
Add 6 to both sides:
[tex]\[ 12x - 6 + 6 \geq 0 + 6 \][/tex]
[tex]\[ 12x \geq 6 \][/tex]
Step 4: Solve for [tex]\(x\)[/tex] by dividing both sides by 12.
[tex]\[ \frac{12x}{12} \geq \frac{6}{12} \][/tex]
[tex]\[ x \geq \frac{1}{2} \][/tex]
Therefore, the solution to the inequality is:
[tex]\[ x \geq \frac{1}{2} \][/tex]
In interval notation, this is expressed as:
[tex]\[ \left[\frac{1}{2}, \infty\right) \][/tex]
So, from the given options, the correct answer is:
[tex]\[ x \geq 0.5 \][/tex]
