A compound inequality includes two inequalities in one statement. A statement such as [latex]4 < x\le 6[/latex] means [latex]4 < x[/latex] and [latex]x\le 6[/latex]. There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.
compound inequality
A compound inequality is a mathematical statement that combines two inequalities into one expression using the words “and” or “or”. These are used to express a range of possible solutions or conditions that satisfy more than one requirement simultaneously.
“And” Compound Inequality (Conjunction):
This type connects two inequalities where both conditions must be true simultaneously.
For example: [latex]a < x < b[/latex]
Or” Compound Inequality (Disjunction)
This type connects two inequalities where at least one of the conditions must be true. It is used to express that solutions may satisfy any one of multiple criteria.
For example: [latex]x < a \text{ or } x > b[/latex]
Solve the compound inequality:
[latex]3\le 2x+2 < 6[/latex]
[latex]\begin{array}{ll} \text{Given inequality:} & 3 \leq 2x + 2 < 6 \\ \text{Subtract 2 from all parts:} & 3 - 2 \leq 2x + 2 - 2 < 6 - 2 \\ \text{Simplify:} & 1 \leq 2x < 4 \\ \text{Divide all parts by 2:} & \frac{1}{2} \leq x < 2 \end{array}[/latex]
In interval notation, the solution is: [latex][\frac{1}{2}, 2)[/latex]
In interval notation, the solution is: [latex](-\infty, -4] \cup (1, \infty)[/latex]
Solve the compound inequality:
[latex]3+x > 7x - 2 > 5x - 10[/latex]
To solve the compound inequality with variables on all 3 parts, we’ll handle it in parts to isolate [latex]x[/latex] in both sections of the inequality. We can follow the steps methodically to ensure each part of the compound inequality is addressed correctly.
[latex]\begin{array}{rcl} 3 + x > 7x - 2 & \quad\text{and}\quad & 7x - 2 > 5x - 10 \\ 3 + x + 2 > 7x -2 +2 & \quad\text{Add 2}\quad & 7x - 2 + 2 > 5x - 10 + 2 \\ 5 + x > 7x & \quad\text{Simplify}\quad & 7x > 5x - 8 \\ 5 > 6x & \quad\text{Isolate } x \text{ to one side}\quad & 2x > -8 \\ \frac{5}{6} > x & \quad\text{Divide by the coefficient of }x \quad & x > -4 \\ x < \frac{5}{6} & \quad\text{Rewrite}\quad & -4 < x \end{array}[/latex]
Combining the two parts: [latex]-4 < x < \frac{5}{6}[/latex]
In interval notation, the solution is: [latex](-4, \frac{5}{6})[/latex]
Number Line:
Notice that when we write the solution in interval notation, the smaller number comes first. We read intervals from left to right as they appear on a number line.
