Solve the inequality 1 - x/3 <= 1 - 2x/7 and show the solution set on a number line.

Step-by-step solution

1. 1

   Cancel the constant that appears on both sides. Subtracting $1$ from each side is legal for an inequality (adding or subtracting never changes the direction) and immediately kills the constants: $-\frac{x}{3}\le-\frac{2x}{7}.$ Clearing denominators first would work too, but cancelling the $1$ keeps the numbers smaller.

2. 2

   Gather the $x$ terms on one side. Add $\dfrac{2x}{7}$ to both sides: $-\frac{x}{3}+\frac{2x}{7}\le 0.$ Collecting on the left rather than the right avoids a needless sign change later.

3. 3

   Combine over the common denominator $21$. Since $\operatorname{lcm}(3,7)=21$, $-\frac{7x}{21}+\frac{6x}{21}\le 0\quad\Longrightarrow\quad -\frac{x}{21}\le 0.$ The two fractions nearly cancel, leaving a single coefficient of $-\tfrac{1}{21}$.

4. 4

   Divide by the negative coefficient and flip the sign. Multiplying both sides by $-21$ reverses the inequality: $-\frac{x}{21}\le 0\quad\Longrightarrow\quad x\ge 0.$ This reversal is the only step where the direction changes, and forgetting it is the classic error - it would give the wrong answer $x\le 0$.

5. 5

   Test values to confirm. At $x=7$: the left side is $1-\tfrac73=-1.3\overline{3}$ and the right side is $1-2=-1$, and $-1.333\le-1$ holds. At $x=-7$: the left side is $1+\tfrac73=3.3\overline{3}$ and the right side is $1+2=3$, and $3.333\le 3$ is false. So non-negative $x$ works and negative $x$ does not, as claimed. At $x=0$ both sides equal $1$, so the endpoint is included.

6. 6

   Describe the number-line picture. Mark a solid (filled) dot at $0$ to show that $0$ itself is a solution, then shade the ray running to the right, towards $+\infty$. In interval notation the solution set is $[0,+\infty)$.