Inequalities with Negatives and Fractions

When the coefficient of $x$ is negative (e.g. $-3x$, $-5x$, $-x$), dividing both sides to isolate $x$ means dividing by a negative number. This always triggers the flip rule.

$-3x < 12$: divide by $-3$ → flip $<$ to $>$: $x > -4$. The classmate in today's hook forgot to flip, giving the wrong direction entirely.

Negative coefficient rule: dividing both sides by a negative number always flips the inequality sign (< → >, ≤ → ≥, etc.). This is the most common error in inequality questions. Check: is the coefficient of x negative? If yes, flip when you divide.

Pause, copy the negative coefficient flip rule (−ax > b → divide by −a → flip → x < −b/a) and the checking question ("is the coefficient of x negative?"), if yes, the flip occurs when you divide by it into your book.

We just saw that negative whole-number coefficients always require a flip when you divide both sides, e.g. −2x > 6 → divide by −2 → flip → x < −3. That raises a question: what about fractional coefficients like x/3 > 4 or −x/2 < 5, do you multiply (not divide), and does the flip rule still apply? This card answers it → multiply both sides by the denominator to clear the fraction; the flip only applies if that denominator is negative.

For $\dfrac{x}{3} < 5$: the coefficient of $x$ is $+\frac{1}{3}$. Multiply both sides by $+3$ (positive) → no flip: $x < 15$.

For $\dfrac{-x}{2} \geq 3$: the coefficient of $x$ is $-\frac{1}{2}$. Multiply both sides by $-2$ (negative) → flip: $x \leq -6$. Or equivalently: multiply by $+2$ to get $-x \geq 6$, then multiply by $-1$ (flip): $x \leq -6$.

Fractional coefficient x/k: multiply both sides by k. If k > 0, no flip. If k < 0 (e.g. −x/2), multiplying by k means multiplying by a negative → flip. Always identify the sign of the coefficient of x before deciding whether to flip.

Pause, copy the fractional coefficient method: multiply both sides by the denominator to clear the fraction, then solve; the flip rule applies only if the multiplier is negative (e.g. −x/2 < 5: multiply by −2 → flip → x > −10) into your book.

We just saw fractional coefficients, multiply by the denominator, and flip only if the denominator is negative. That raises a question: a mixed problem has both fractions and a negative coefficient on x, what order do you tackle the steps to avoid a sign error? This card answers it → isolate the variable term first (add/subtract constants from both sides), then deal with the coefficient (multiply to clear fractions, check sign → flip or no flip).

Multi-step problems may involve both a negative coefficient and a fraction. Isolate the variable term first (add/subtract), then deal with the coefficient. Apply the flip rule exactly once, only when the final division/multiplication is by a negative.

Example: $-4x + 6 > -10 \Rightarrow -4x > -16 \Rightarrow x < 4$ (divide by $-4$, flip $>$ to $<$).

Multi-step inequality with both negative coefficient and fractions: isolate the variable term first (add/subtract to remove constants), then deal with the coefficient (multiply/divide, check sign for flip). Work one step at a time; show each operation explicitly.

Pause, copy the multi-step inequality strategy: (1) isolate the variable term by adding/subtracting constants first; (2) then multiply or divide by the coefficient, checking its sign; (3) if the coefficient is negative, flip, this order prevents sign errors into your book.

1. Solve $-6x > 24$. Show the flip step clearly.
2. Solve $\dfrac{x}{5} \leq 4$.
3. Solve $-\dfrac{x}{3} < 2$. Identify where the flip occurs.
4. A classmate solved $-2x > 8$ and wrote $x > -4$. Identify and correct the error.