Solving a one-step linear inequality with subtraction

Key concept: This question checks whether you can solve a linear inequality by isolating the variable correctly.

Step by step

Start with the inequality

We are given:

$1 - 2x \le 3$

The goal is to isolate $x$. Begin by subtracting 1 from both sides:

$-2x \le 2$

Now divide both sides by -2. Because you are dividing by a negative number, the inequality sign must reverse:

$x \ge -1$

So the correct answer is (d) $x \ge -1$.

Why the sign flips

This is the most important rule in inequality solving. When you multiply or divide by a negative number, the inequality direction changes. That happens because negative scaling reverses the order of numbers on the number line.

If you skipped the sign change, you would incorrectly write $x \le -1$, which is a common error.

Check by substitution

Test the boundary value $x = -1$ in the original inequality:

$1 - 2(-1) = 1 + 2 = 3$

Since $3 \le 3$ is true, the endpoint works. Now try a value larger than -1, like 0:

$1 - 2(0) = 1 \le 3$

That is also true. Try a value less than -1, like -2:

$1 - 2(-2) = 5 \le 3$

That is false. This confirms that all solutions are numbers greater than or equal to -1.

Key rule to remember

For equations, dividing by a negative is routine. For inequalities, dividing by a negative always reverses the symbol. That one rule decides this problem.

Pitfall alert

A very common mistake is to stop after getting -2x ≤ 2 and then divide by -2 without flipping the inequality sign. That leads to the wrong answer. Another error is to move 1 to the right side as if it were an equation but forget that the original symbol is still an inequality. When the coefficient of x is negative, always check whether the sign must reverse.

Try different conditions

If the inequality were $1 - 2x < 3$, the solution would be $x > -1$ instead of $x \ge -1$ because the strict inequality sign must also flip when dividing by a negative number. If the problem were $1 + 2x \le 3$, then after subtracting 1 you would get $2x \le 2$, and dividing by positive 2 would give $x \le 1$ without reversing the sign.

Further reading

inequality sign reversal, isolating variables, negative coefficient