Solve the inequality 3|x−1| > |2x+4|

Full worked solution

1. Eliminate absolute value signs by squaring both sides

   Since both sides of the inequality are non-negative, we use the identity $|a{|}^2=a^2$ for all real $a$ to square both sides without changing the direction of the inequality:

   $$(3|x-1|)^2 > |2x+4|^2$$

   Expanding both sides:

   $$9(x-1)^2 > (2x+4)^2$$

   Expand the squared terms:

   $$9(x^2 - 2x + 1) > 4x^2 + 16x + 16$$

   Rearrange all terms to one side to form a standard quadratic inequality:

   $$9x^2 - 18x + 9 - 4x^2 - 16x - 16 > 0$$

   Simplify:

   $$5x^2 - 34x - 7 > 0$$

   Note.

   We can only square both sides safely because both sides are non-negative. Never square both sides of an inequality where one or both sides could be negative, as this can reverse the inequality direction unpredictably.

2. Find critical values by solving the quadratic equation

   First solve the corresponding quadratic equation $5x^2-34x-7=0$ using the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, where $a=5$, $b=-34$, $c=-7$. Calculate the discriminant:

   $$\Delta = b^2 - 4ac = (-34)^2 - 4(5)(-7) = 1156 + 140 = 1296 = 36^2$$

   Substitute back into the quadratic formula:

   $$x = \frac{34 \pm 36}{2(5)} = \frac{34 \pm 36}{10}$$

   Compute the two roots:

   • First root: $x=\frac{34+36}{10}=\frac{70}{10}=7$
   • Second root: $x=\frac{34-36}{10}=\frac{-2}{10}=-\frac{1}{5}$

   These roots $x=-\frac{1}{5}$ and $x=7$ are the critical values where the quadratic equals zero.

   Note.

   Double-check the sign of $b$ in the quadratic formula: since $b=-34$, $-b=+34$, which we used correctly above. Mixing up the sign of $b$ is a common source of error here.

3. Determine the solution interval from the parabola shape

   The quadratic $5x^2-34x-7$ has a positive leading coefficient ($a=5>0$), so its graph is an upward-opening parabola that tends to $+\infty$ as $x\to \pm \infty$. It is greater than zero outside of its two roots. We verify this with a test point in each region:

   • For $x<-\frac{1}{5}$, pick $x=-1$: $5(-1{)}^2-34(-1)-7=5+34-7=32>0$, which satisfies the inequality.
   • For $-\frac{1}{5}<x<7$, pick $x=0$: $5(0)-34(0)-7=-7<0$, which does not satisfy the inequality.
   • For $x>7$, pick $x=8$: $5(64)-34(8)-7=320-272-7=41>0$, which satisfies the inequality.

   To visualize this, we sketch the quadratic and the two sides of the original inequality:

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