Two-step equations and inequalities Knowledge-Based Study

Study notes

1. What is a Two-Step Equation?

   A two-step equation is an algebraic equation that requires exactly two inverse operations to isolate the variable. It typically appears in the standard linear form:

   $$ax + b = c$$

   where a, b, and c are known constants and x represents the unknown variable.

   The Order of Inverse Operations

   To isolate the variable, undo operations in the reverse order of the standard order of operations (PEMDAS):

   1. Step 1: Undo addition or subtraction first by applying the opposite operation to both sides of the equation.
   2. Step 2: Undo multiplication or division by multiplying or dividing both sides by the coefficient of the variable.

   Worked Example

   Solve for x: $3x + 7 = 22$

   • Step 1 (Undo addition): Subtract 7 from both sides.
     $3x + 7 - 7 = 22 - 7$
     $3x = 15$
   • Step 2 (Undo multiplication): Divide both sides by 3.
     $\frac{3x}{3} = \frac{15}{3}$
     $x = 5$

   Always verify by substituting $x = 5$ back into the original equation: $3(5) + 7 = 15 + 7 = 22$, which is true.

   Two-Step Equations Involving Division

   Equations can also feature variables divided by a constant, written in the form:

   $$\frac{x}{a} - b = c$$

   Worked Example 1: Isolating a Divided Variable

   Solve for y: $\frac{y}{4} - 5 = 3$

   1. Step 1: Add 5 to both sides to eliminate the constant term.
      $\frac{y}{4} - 5 + 5 = 3 + 5$
      $\frac{y}{4} = 8$
   2. Step 2: Multiply both sides by 4 to clear the denominator.
      $4 \cdot \frac{y}{4} = 4 \cdot 8$
      $y = 32$

   Worked Example 2: Negative Coefficients

   When the coefficient is negative, pay close attention to signs:

   Solve for m: $-2m + 9 = -1$

   • Subtract 9 from both sides: $-2m = -10$
   • Divide both sides by -2: $m = \frac{-10}{-2} = 5$

   Understanding Two-Step Inequalities

   An inequality shows that two values are not necessarily equal, using relational symbols:

   • $<$ (less than)
   • $>$ (greater than)
   • $\le$ (less than or equal to)
   • $\ge$ (greater than or equal to)

   Unlike linear equations, which typically have a single unique solution, inequalities have an entire set of solutions that make the statement true.

   Solving Form $ax + b > c$

   The general procedure mirrors solving two-step equations: isolate the variable term first, then isolate the variable itself.

   Worked Example

   Solve for x: $4x - 3 \le 13$

   1. Add 3 to both sides:
      $4x - 3 + 3 \le 13 + 3$
      $4x \le 16$
   2. Divide both sides by 4:
      $x \le 4$

   Any real number less than or equal to 4 satisfies this inequality.

   The Fundamental Rule of Inequalities

   When solving an inequality, there is one crucial distinction from equations:

   Whenever you multiply or divide both sides of an inequality by a negative number, you must reverse (flip) the inequality symbol.

   Why Does the Sign Reverse?

   Consider the true numerical statement $2 < 5$. If you multiply both sides by -1:

   $$-2 \text{ versus } -5$$

   On a number line, -2 is to the right of -5, meaning $-2 > -5$. The relationship inverts.

   Worked Example

   Solve for k: $-5k + 6 < 26$

   1. Step 1: Subtract 6 from both sides.
      $-5k < 20$
   2. Step 2: Divide both sides by -5 and reverse the inequality sign from $<$ to $>$.
      $\frac{-5k}{-5} > \frac{20}{-5}$
      $k > -4$

   Checking the Solution

   Select a test value greater than -4, such as $k = 0$:
   $-5(0) + 6 = 6 < 26$, which holds true.

   Two-Step Solving Strategy

   Follow this checklist when solving two-step algebraic sentences:

   1. Identify the operations: Determine what is currently being done to the variable (e.g., multiplied by 3 and added to 4).
   2. Undo addition and subtraction: Apply the inverse operation to both sides of the equal sign or inequality symbol.
   3. Undo multiplication and division: Apply the inverse operation to isolate the variable.
   4. Check for sign reversal (Inequalities only): If you multiply or divide by a negative value, immediately reverse the direction of the inequality sign.
   5. Verify: Substitute your solution (or a sample test point from your solution set) back into the original expression to confirm accuracy.

   Common Pitfalls to Avoid

   • Dividing only one term instead of the entire opposite side.
   • Reversing the inequality symbol when subtracting a number (reversal occurs only when multiplying or dividing by a negative).
   • Dropping negative signs attached to variable coefficients.