A double inequality is a specific type of compound inequality, written continuously in the form $$a < x < b$$. This concise mathematical notation represents two simultaneous conditions joined by the logical operator "and", meaning it is completely equivalent to stating that $$a < x$$ and $$x < b$$. Double inequalities can also use other combinations of strict and non-strict symbols, such as $$a \leq x \leq b$$, $$a > x > b$$, or $$a \geq x \geq b$$. Each of these continuous forms translates to an "and" relationship between the two boundaries, effectively describing a single, bounded interval on the number line.

Double Inequality

To solve a compound inequality means to find all values of the variable that make the compound statement true. The process uses the same algebraic techniques as solving single linear inequalities. Each inequality is solved separately, and then the two resulting solutions are considered together to find the final values that satisfy the specific "and" or "or" condition.

Claude

A compound inequality is formed when two distinct inequalities are logically connected by the word "and" or the word "or". This mathematical structure combines multiple conditions that must be evaluated simultaneously to determine the complete encompassing solution set.

Compound Inequality

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Solving Compound Inequalities

When evaluating a compound inequality connected by the word "and", the solution consists of all numbers that make both inequalities true simultaneously. This shared set of solutions is known as the intersection of the two inequalities. Geometrically, if you graph each inequality on a number line, the intersection is the specific region where the shaded parts of both graphs overlap.

Intersection of Two Inequalities

To visually determine the solution for the compound inequality $$x > -3$$ and $$x \leq 2$$, graph each inequality on a number line and locate the intersection. The graph of $$x > -3$$ extends to the right of $$-3$$ (not including $$-3$$), while the graph of $$x \leq 2$$ extends to the left of $$2$$ (including $$2$$). The overlapping region—where numbers are shaded on both individual graphs—forms the solution. Because $$-3$$ is not shaded on the first graph, it is excluded from the solution. The number $$2$$ is shaded on both, so it is included. The final solution spans the interval strictly between $$-3$$ and up to and including $$2$$.

Example: Graphing the Intersection of $$x > -3$$ and $$x \leq 2$$

To systematically solve a compound inequality with the word "and", follow a three-step procedure. Step 1 involves independently solving each of the individual linear inequalities. Step 2 requires graphing each solution on a number line, and then graphing the numbers that make both inequalities true by finding where the initial graphs overlap. Step 3 consists of writing this final overlapping solution utilizing standard interval notation.

Procedure for Solving Compound Inequalities with "and"

To solve the compound inequality $$6x - 3 < 9$$ and $$2x + 9 \geq 3$$, begin by solving each inequality. For the first, adding $$3$$ gives $$6x < 12$$, which simplifies to $$x < 2$$. For the second, subtracting $$9$$ gives $$2x \geq -6$$, which simplifies to $$x \geq -3$$. Graphing $$x < 2$$ results in shading to the left of an open parenthesis at $$2$$, while graphing $$x \geq -3$$ results in shading to the right of a bracket at $$-3$$. The final solution is the overlapping region, which spans from $$-3$$ up to $$2$$. In interval notation, this is written as $$[-3, 2)$$.

Example: Solving and Graphing $$6x - 3 < 9$$ and $$2x + 9 \geq 3$$

To practice solving a compound inequality with "and", solve the inequalities $$4x - 7 < 9$$ and $$5x + 8 \geq 3$$. Solving the first inequality gives $$4x < 16$$, which simplifies to $$x < 4$$. Solving the second inequality yields $$5x \geq -5$$, which simplifies to $$x \geq -1$$. Graphing both results shows that the intersection—the numbers making both true—is between $$-1$$ and $$4$$. The boundary $$-1$$ is included while $$4$$ is not. In interval notation, this overlapping solution is written as $$[-1, 4)$$.

Try It: Solving and Graphing $$4x - 7 < 9$$ and $$5x + 8 \geq 3$$

To further practice solving "and" compound inequalities, consider $$3x - 4 < 5$$ and $$4x + 9 \geq 1$$. For the first inequality, adding $$4$$ gives $$3x < 9$$, so $$x < 3$$. For the second, subtracting $$9$$ gives $$4x \geq -8$$, so $$x \geq -2$$. The numbers that make both inequalities true are those where the graphs of $$x < 3$$ and $$x \geq -2$$ overlap. This interval includes $$-2$$ but excludes $$3$$. Written in interval notation, the final solution is $$[-2, 3)$$.

Try It: Solving and Graphing $$3x - 4 < 5$$ and $$4x + 9 \geq 1$$

When evaluating a compound inequality connected by the word 'or', the solution consists of all numbers that make either inequality true. This combined set of solutions is the union of the two inequalities. Geometrically, if you graph each inequality on a number line, the union is the entire region formed by combining the shaded parts of both individual graphs. Just as the United States is the union of all 50 states, the mathematical solution is the union of all numbers that belong to either graph.

Union of Two Inequalities

For any algebraic expression $$u$$ and any positive real number $$a$$, the absolute value inequality $$|u| < a$$ can be translated into the compound inequality $$-a < u < a$$. Similarly, the inequality $$|u| \leq a$$ translates to $$-a \leq u \leq a$$. This algebraic property demonstrates that the solution set naturally includes all values whose underlying distance from zero is constrained to be strictly less than (or less than or equal to) $$a$$. Geometrically, this solution corresponds precisely to the continuous segment on the number line bounded securely between $$-a$$ and $$a$$.

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