3.3 Solving Linear Inequalities

Understanding Linear Inequalities

A linear inequality in one variable is a mathematical statement comparing two algebraic expressions using an inequality symbol rather than an equals sign. Unlike linear equations, which typically yield a single discrete numerical solution, linear inequalities yield a range or set of infinitely many solutions that satisfy the given condition.

In healthcare and clinical nursing practice, vital signs, lab values, and medication safety parameters are almost exclusively defined as inequality ranges. For instance, maintaining blood glucose within a target range ($70 \le G \le 99\text{ mg/dL}$), ensuring maximum daily Acetaminophen dose does not exceed $4,000\text{ mg}$ ($D \le 4000$), or monitoring pediatric urine output thresholds ($U \ge 1\text{ mL/kg/hr}$) all rely directly on inequality logic.

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Inequality Symbols, Number Line Graphs, and Interval Notation

Mastering the four primary inequality symbols, their corresponding verbal descriptions, number line representations, and formal interval notation is essential for TEAS 7 success.

Key Concept — Open vs. Closed Endpoints: An open circle or parenthesis () indicates that the boundary value is excluded from the solution set. A closed circle or square bracket [] indicates that the boundary value is included in the solution set. Infinity symbols ($-\infty, \infty$) always take parentheses ().

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The Golden Rule of Inequalities: The Negative Flip Rule

Solving linear inequalities follows almost the exact same step-by-step procedures as solving linear equations, with one vital exception:

CRITICAL RULE: Whenever you multiply or divide both sides of an inequality by a negative number, you MUST reverse (flip) the direction of the inequality symbol ($<$ becomes $>$, $>$ becomes $<$, $\le$ becomes $\ge$, and $\ge$ becomes $\le$).

Why Does the Inequality Symbol Flip?

Consider the true numerical inequality: $2 < 5$.

• Multiply both sides by $+2$: $4 < 10$ (STILL TRUE).
• Multiply both sides by $-1$: $-2$ and $-5$. On a number line, $-2$ lies to the right of $-5$, meaning $-2 > -5$. To maintain truth, the sign must flip!

Step-by-Step Worked Example 1: Basic Negative Division

Solve the inequality for $x$: $-4x + 9 \ge 29$

• Step 1 (Subtract 9 from both sides): $-4x \ge 29 - 9 \implies -4x \ge 20$
• Step 2 (Divide both sides by -4 AND FLIP THE INEQUALITY SIGN): $\frac{-4x}{-4} \le \frac{20}{-4} \implies x \le -5$
• Step 3 (Graph & Interval Notation): Closed circle at $-5$, shading to the left. Interval: $(-\infty, -5]$.

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Multi-Step Inequalities with Variables on Both Sides

When variables appear on both sides of an inequality, collect variable terms on one side. You can avoid dividing by negative coefficients altogether by moving the variable to the side that yields a positive coefficient.

Step-by-Step Worked Example 2: Multi-Step Inequality

Solve for $x$: $3(x - 4) - 2(x + 1) < 4x + 1$

• Step 1 (Distribute parentheses): $3x - 12 - 2x - 2 < 4x + 1$
• Step 2 (Combine like terms on left side): $(3x - 2x) + (-12 - 2) < 4x + 1 \implies x - 14 < 4x + 1$
• Step 3 (Method A — Move variable to left):
  • Subtract $4x$ from both sides: $-3x - 14 < 1$
  • Add $14$ to both sides: $-3x < 15$
  • Divide by $-3$ and flip sign: $x > -5$
• Step 4 (Method B — Move variable to right to stay positive):
  • Subtract $x$ from both sides: $-14 < 3x + 1$
  • Subtract $1$ from both sides: $-15 < 3x$
  • Divide by $+3$ (no sign flip): $-5 < x$, which is identical to $x > -5$.
• Result: $x > -5$ or in interval notation: $(-5, \infty)$.

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Healthcare Application: Safe Maximum Daily Dosage

In clinical pharmacology, maximum daily safety caps are expressed as inequalities to prevent drug toxicity.

Clinical Problem: Maximum Daily Tylenol Ceiling

A pediatric patient weighing $20\text{ kg}$ is prescribed Acetaminophen (Tylenol). The safe total daily dosage $D$ in milligrams must satisfy the compound inequality:

$30W + 150 \le D \le 45W + 200$

where $W$ is the child's weight in kilograms.

• Task: Determine the maximum safe total daily dosage $D$ for this $20\text{ kg}$ child.
• Step 1 (Identify the upper bound expression): $D \le 45W + 200$
• Step 2 (Substitute $W = 20\text{ kg}$): $D \le 45(20) + 200$
• Step 3 (Calculate product): $45 \cdot 20 = 900$
• Step 4 (Calculate sum): $D \le 900 + 200 \implies D \le 1,100\text{ mg/day}$
• Clinical Conclusion: The nurse must ensure the child receives no more than $1,100\text{ mg}$ in a 24-hour period.

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TEAS 4-Function Calculator Strategy

1. Testing Boundary Points: Use the 4-function calculator to verify your boundary number. For $x > -5$, test $x = -5$ in the original equation to ensure both sides equal each other.
2. Testing Solution Values: Pick a simple test number inside your solution set (e.g., for $x > -5$, choose $x = 0$).
   • Plug $x = 0$ into $3(0 - 4) - 2(0 + 1) < 4(0) + 1$.
   • Left side: $3(-4) - 2(1) = -12 - 2 = -14$.
   • Right side: $4(0) + 1 = 1$.
   • Statement: $-14 < 1$ (TRUE!). This confirms your inequality direction is correct.

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Common Student Traps & How to Avoid Them

1. Forgetting to Flip on Negative Division: Writing $-3x > 15 \implies x > -5$ (WRONG). Remedy: Highlight every negative coefficient in red on your scratch paper as a reminder to flip the sign.
2. False Flip on Negative Numerators: Dividing a negative numerator by a positive denominator, such as $2x < -10 \implies x > -5$ (WRONG). Remedy: Remember sign flips ONLY occur when the divisor (bottom number) is negative! Since $+2$ is positive, the sign stays: $x < -5$.
3. Misinterpreting Verbal Phrases: Translating 'at least 15' as $x < 15$. Remedy: 'At least 15' means 15 or more ($x \ge 15$). 'At most 15' means 15 or less ($x \le 15$).