Question

Solve and graph. $$ 12-y \geq 2(9-2 y) $$

   Solve and graph.
$$
12-y \geq 2(9-2 y)
$$

Precalculus

Raymond A. Barnett,… 7th Edition

Chapter 1, Problem 31

Instant Answer

Solved by Expert Nick Johnson

07/20/2021

Step 1

Step 1: First, distribute the 2 in the right side of the inequality: $$ 12-y \geq 2(9-2y) \Rightarrow 12-y \geq 18-4y $$ Show more…

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Solve and graph. $$ 12-y \geq 2(9-2 y) $$

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Key Concepts

Linear Inequalities

A linear inequality is an expression that relates two linear expressions using inequality symbols (like ?, ?, >, or <) instead of an equal sign. Solving these involves finding all values of the variable that make the inequality true. The process is similar to solving linear equations, but special care must be taken when performing operations, such as multiplying or dividing by a negative number, which reverses the inequality sign.

Properties of Inequalities

The properties of inequalities are similar to those of equations, particularly the addition and subtraction properties which state that the same value can be added or subtracted from both sides without affecting the inequality. However, when multiplying or dividing both sides by a negative number, the inequality sign must be flipped. These properties help in isolating the variable and developing the solution set.

Graphing Solutions on the Number Line

Graphing solutions on a number line involves representing all the numbers that satisfy an inequality. For inclusive inequalities (like ? or ?), a closed circle is used at the boundary point to indicate that the endpoint is included in the solution set. The portion of the number line that satisfies the inequality is then shaded. This visual representation helps in understanding the range of possible solutions.

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