Linear equations
Do the same thing to both sides until $x$ is alone. With fractions, multiply both sides by the lowest common denominator first to clear them.
Solve $\dfrac{x + 1}{3} - \dfrac{x - 2}{4} = 1$.
$4(x + 1) - 3(x - 2) = 12$
Multiply every term by 12, the LCD of 3 and 4.
$4x + 4 - 3x + 6 = 12$
Expand. −3 × −2 = +6.
$x + 10 = 12$, so $x = 2$.
Linear inequations
An is solved like an equation, with one extra rule.
Solve $5 - 3x \ge 11$ and show the solution on a number line.
$x \le -2$
Dividing by −3, a negative, so the sign reverses.
On the number line: a filled dot at $-2$ with an arrow to the left.
The green band shows the x-values where the line is on or above the x-axis. Try a negative m and watch which side gets shaded.
Example 3: a double inequation●●●Solve $-4 < 2 - 3x \le 8$.
$-6 < -3x \le 6$
Do the same to all three parts: subtract 2.
$2 > x \ge -2$
Divide all parts by −3 and reverse both signs.
$-2 \le x < 2$, or $[-2, 2)$.
Rewrite with the smaller number first.
When does the inequality sign flip?
When you multiply or divide both sides by a negative.
Fractions in an equation
Multiply every term by the LCD first.
Double inequation
Do the same operation to all three parts.
Solve $-4x > 12$.
Divide by $-4$ and reverse: $x < -3$.
Solve $\dfrac{x}{2} + \dfrac{x}{5} = \dfrac{49}{10}$.
Multiply by 10: $5x + 2x = 49$, so $x = 7$.
Solve $3 \le 2x + 1 < 9$.
Subtract 1: $2 \le 2x < 8$. Divide by 2: $1 \le x < 4$.
Solve $2(x - 3) = x + 4$.
$2x - 6 = x + 4$, so $x = 10$.
Solve $7 - 2x < 1$.
$-2x < -6$. Divide by $-2$ and reverse: $x > 3$.
Solve $\dfrac{x - 4}{5} = \dfrac{x + 2}{3}$.
Multiply by $15$: $3x - 12 = 5x + 10$, so $-2x = 22$ and $x = -11$.
Which step reverses an inequality sign?
Only multiplying or dividing by a negative reverses the sign.
Solve $\dfrac{x}{3} - 1 \ge 2$.
$\dfrac{x}{3} \ge 3$, so $x \ge 9$.
New numbers every time. Keep going until you get five in a row.
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