How to solve compound inequalities

Solving compound inequalities involves finding the values of a variable that satisfy multiple inequalities simultaneously. Compound inequalities are typically connected by “and” (conjunction, where both conditions must be true) or “or” (disjunction, where at least one condition must be true). Here’s a clear, step-by-step guide to solve them, with examples for both types.

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General Steps for Solving Compound Inequalities

1. Identify the Type of Compound Inequality:
   • “And” Inequality: Both conditions must be satisfied (e.g., $-3<x<5$).
   • “Or” Inequality: At least one condition must be satisfied (e.g., $x<-2$ or $x>3$).
2. Break It Down:
   • For “and” inequalities, solve each part separately or treat them as a single expression if written compactly (e.g., $a<x<b$).
   • For “or” inequalities, solve each inequality independently.
3. Solve Each Inequality:
   • Use standard inequality-solving techniques (similar to solving equations):
     • Isolate the variable by adding, subtracting, multiplying, or dividing both sides.
     • Reverse the inequality sign if multiplying or dividing by a negative number.
   • Simplify to find the solution for each part.
4. Combine the Solutions:
   • For “and”: Find the intersection of the solutions (where both conditions overlap).
   • For “or”: Find the union of the solutions (all values that satisfy at least one condition).
5. Graph the Solution (Optional):
   • Plot the solution on a number line to visualize the range of values.
   • Use closed dots ($\bullet$) for $\le$ or $\ge$, and open dots ($\circ$) for $<$ or $>$.
   • Shade the appropriate regions (overlapping for “and,” separate for “or”).
6. Write the Solution:
   • Express in interval notation or inequality notation, depending on the requirement.
   • Check for special cases (e.g., no solution or all real numbers).

Example 1: “And” Compound Inequality

Solve: $-2<3x+1<10$

1. Treat as a Single Expression:
   • Since it’s an “and” inequality, solve the entire expression at once.
   • Start with: $-2<3x+1<10$.
2. Isolate the Variable:
   • Subtract 1 from all parts: $-2-1<3x+1-1<10-1$ $-3<3x<9$.
   • Divide all parts by 3: −33<3×3<93 \frac{-3}{3} < \frac{3x}{3} < \frac{9}{3} 3−3​<33x​<39​ $-1<x<3$.
3. Solution:
   • Inequality notation: $-1<x<3$.
   • Interval notation: $(-1,3)$.
   • Graph: On a number line, shade between -1 and 3 with open dots at -1 and 3 (since $<$).
4. Check:
   • Test $x=0$: $-2<3(0)+1<10\to -2<1<10$, which is true.

Example 2: “Or” Compound Inequality

Solve: $2x-3\le 5$ or $x+1>4$

1. Solve Each Inequality Separately:
   • First inequality: $2x-3\le 5$
     • Add 3: $2x\le 8$.
     • Divide by 2: $x\le 4$.
   • Second inequality: $x+1>4$
     • Subtract 1: $x>3$.
2. Combine Solutions:
   • For “or,” take the union: $x\le 4$ or $x>3$.
   • On a number line, this includes all values $x\le 4$ (from negative infinity to 4) and all values $x>3$ (from 3 to positive infinity).
   • Since the regions overlap at $x>3$, the solution is all real numbers.
3. Solution:
   • Inequality notation: All real numbers.
   • Interval notation: $(-\infty ,\infty )$.
   • Graph: The entire number line is shaded.
4. Check:
   • Test $x=0$: $2(0)-3=-3\le 5$ (true), so it satisfies the first part.
   • Test $x=5$: $5+1=6>4$ (true), so it satisfies the second part.

Example 3: “And” Inequality Written Separately

Solve: $2x+1>3$ and $x-4\le 2$

1. Solve Each Inequality:
   • First: $2x+1>3$
     • Subtract 1: $2x>2$.
     • Divide by 2: $x>1$.
   • Second: $x-4\le 2$
     • Add 4: $x\le 6$.
2. Combine Solutions:
   • For “and,” take the intersection: $x>1$ and $x\le 6$.
   • This means $1<x\le 6$.
3. Solution:
   • Inequality notation: $1<x\le 6$.
   • Interval notation: $(1,6]$.
   • Graph: Shade between 1 and 6, with an open dot at 1 and a closed dot at 6.
4. Check:
   • Test $x=2$: $2(2)+1=5>3$ (true) and $2-4=-2\le 2$ (true).

Special Cases

• No Solution: If the conditions can’t overlap (e.g., $x<2$ and $x>5$), the solution is empty ($\varnothing$).
• All Real Numbers: If the solution covers all numbers (as in Example 2), use $(-\infty ,\infty )$.
• Negative Coefficients: If you multiply/divide by a negative number, flip the inequality sign. Example: Solve $-2x>4$. Divide by -2: $x<-2$.

Tips

• Always double-check by testing a value in the solution range.
• If graphing, ensure the correct use of open/closed dots based on $<$ vs. $\le$.
• For complex inequalities, write each step clearly to avoid mistakes.

If you have a specific compound inequality to solve or want a visual graph (I can provide a description or guide you to plot it), let me know!

1. Example 1: Solve the compound inequality -7 < -3x – 2 ≤ 5 and represent the solution in the interval notation.

   Solution:

   The given inequality is,

   -7 < -3x – 2 ≤ 5

   Adding 2 on all the sides,

   -5 < -3x ≤ 7

   Dividing all the sides by -3 (note that the signs of inequalities change as we are dividing by negative number),

   5/3 > x ≥ -7/3

   This can be written as -7/3 ≤ x < 5/3.

   Hence, the solution in the interval notation is [-7/3, 5/3).

   Answer: [-7/3, 5/3)