One-Variable Inequalities

Learn inequality solution sets, reversing an inequality.

Lesson 3 of 32 parts Practice Flashcards

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Inequality Solution Sets

Objective: Explain inequality solution sets, apply the stated method, and check its conditions.

A one-variable inequality identifies a set of values. Adding or subtracting the same quantity preserves direction; dividing by a positive constant also preserves direction. A strict inequality excludes its boundary value, shown by an open number-line endpoint. A strict inequality uses an open endpoint.

Formulas

$ax+b<d, a>0 ⇒ x<(d−b)/a$

Worked examples

Try itSolve 6x+7<49. Give its excluded boundary value.Show answer

Step 1: Subtract 7: 6x<42. Step 2: Divide by positive 6: x<7. Step 3: The boundary 7 is excluded.

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Inequality Solution Sets

Objective: Explain inequality solution sets, apply the stated method, and check its conditions.

Formulas

$ax+b<d, a>0 ⇒ x<(d−b)/a$

Worked examples

Try itSolve 6x+7<49. Give its excluded boundary value.Show answer

Step 1: Subtract 7: 6x<42. Step 2: Divide by positive 6: x<7. Step 3: The boundary 7 is excluded.

Reversing an Inequality

Objective: Explain reversing an inequality, apply the stated method, and check its conditions.

Multiplying or dividing an inequality by a negative constant reverses the comparison. This follows because reflection across zero reverses number-line order. Preserve strict versus inclusive status when changing direction, and test a candidate to confirm the resulting interval. Reverse direction for negative multiplication or division.

Formulas

$If a<0, ax<d ⇒ x>d/a$

Worked examples

Try itSolve −6x+7<-35. Give the excluded boundary value.Show answer

Step 1: Subtract 7: −6x<−42. Step 2: Divide by −6 and reverse the sign: x>7. Step 3: The boundary 7 is excluded.

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