Solve the inequality 7x - 15 >= 4x + 6.

Step-by-step solution

1. 1

   Gather the x terms on the side with the larger coefficient. Subtract $4x$ from both sides. Subtraction never changes an inequality's direction, and choosing the smaller coefficient to remove keeps what remains positive:

   $7x-15-4x\ge 4x+6-4x\qquad\Longrightarrow\qquad 3x-15\ge6$

2. 2

   Move the constant across. Add $15$ to both sides:

   $3x\ge21$

3. 3

   Divide by 3, and note why nothing flips. The divisor $3$ is positive, so the $\ge$ survives intact:

   $x\ge7$

   Had the earlier step left $-3x$, dividing by $-3$ would have reversed the sign — which is exactly why step 1 was arranged to avoid that.

4. 4

   Check the boundary. At $x=7$: left $=49-15=34$ and right $=28+6=34$ — equal, so the boundary satisfies the non-strict $\ge$ and is included.

5. 5

   Test a point on each side. At $x=10$ (inside): $70-15=55$ against $40+6=46$, and $55\ge46$ ✓. At $x=0$ (outside): $-15$ against $6$, and $-15\ge6$ is false ✓. So the solution set is exactly $[7,\infty)$.

6. 6

   Verify by scanning. Comparing both sides with exact fractions at $4001$ points on $[-20,20]$, the inequality holds at precisely the points with $x\ge7$ ✓.