Question $7-12 x \geqslant 25-3 x$

Studdy Solution

STEP 1

1. The inequality $7 - 12x \geqslant 25 - 3x$ is a linear inequality.
2. We are solving for the variable $x$ .
3. The inequality involves basic algebraic operations, specifically addition and subtraction.

STEP 2

1. Simplify the inequality by moving terms involving $x$ to one side.
2. Move constant terms to the other side.
3. Solve for $x$ .
4. Check the solution by substituting back into the original inequality.

STEP 3

Start by moving all terms involving $x$ to one side of the inequality. We can do this by adding $12x$ to both sides:
$7 - 12x + 12x \geqslant 25 - 3x + 12x$
This simplifies to:
$7 \geqslant 25 + 9x$

STEP 4

Next, move the constant term $25$ to the other side by subtracting $25$ from both sides:
$7 - 25 \geqslant 9x$
Simplify the left side:
$-18 \geqslant 9x$

STEP 5

Now, solve for $x$ by dividing both sides by $9$ :
$\frac{-18}{9} \geqslant x$
Simplify the fraction:
$-2 \geqslant x$
Or equivalently:
$x \leqslant -2$

STEP 6

Check the solution by substituting $x = -2$ back into the original inequality to ensure it holds:
Original inequality:
$7 - 12(-2) \geqslant 25 - 3(-2)$
Calculate each side:
Left side: $7 + 24 = 31$
Right side: $25 + 6 = 31$
Since $31 \geqslant 31$ is true, the solution is verified.
The solution is:
$x \leqslant -2$

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