Solving inequalities
Solving Inequalities covers the rules for finding all values of a variable that satisfy an inequality, including the critical rule that multiplying or dividing both sides by a negative number reverses the inequality sign. Taught in Yoshiwara Elementary Algebra Chapter 2: Linear Equations, Grade 6 students learn to apply the same steps as equation-solving, with this one key difference. Solutions to inequalities represent an entire range of values and are often expressed in interval notation or on a number line.
Key Concepts
Property To solve an inequality: 1. We can add or subtract the same quantity on both sides. 2. We can multiply or divide both sides by the same positive number. 3. If we multiply or divide both sides by a negative number, we must reverse the direction of the inequality.
Examples To solve $2x 5 < 9$, add 5 to both sides to get $2x < 14$. Then, divide by 2 to get $x < 7$. The inequality sign does not change. To solve $14 4x \geq 6$, subtract 14 to get $ 4x \geq 8$. Then, divide by $ 4$ and reverse the inequality sign to get $x \leq 2$. To solve $2 \frac{x}{3} 4$, subtract 2 to get $ \frac{x}{3} 2$. Then, multiply by $ 3$ and reverse the inequality sign to get $x < 6$.
Explanation Solving an inequality is just like solving an equation, with one crucial exception. Remember this golden rule: if you multiply or divide both sides by a negative number, you MUST flip the direction of the inequality sign ($<$ becomes $ $, and vice versa).
Common Questions
How do you solve an inequality?
Use the same steps as solving an equation — add, subtract, multiply, or divide both sides — but remember to flip the inequality sign whenever you multiply or divide by a negative number.
Why does the inequality sign flip when dividing by a negative?
Because multiplying or dividing by a negative reverses the order of numbers. For example, 2 < 4, but multiplying both by -1 gives -2 > -4.
How do you represent the solution to an inequality?
Write the solution as an inequality (x > 3), plot it on a number line with an open or closed circle and an arrow, or express it in interval notation (3, ∞).
Where are solving inequalities covered in Yoshiwara Elementary Algebra?
Solving inequalities is in Chapter 2: Linear Equations of Yoshiwara Elementary Algebra.
What is the difference between strict and non-strict inequalities?
Strict inequalities (< and >) exclude the boundary value (open circle on number line). Non-strict inequalities (≤ and ≥) include the boundary value (closed circle).