Question

Multiple Choice Solve \(-3 y+2<17\). (Algebra Revietw) (A) \(y<-5\) (B) \(y>18\)

Short answer

Option (B) is correct: \(y > -5\).

Step-by-step solution

Isolate the Variable on One Side

Start by isolating the variable \(y\) on one side of the inequality. Here, you first want to eliminate the constant term \(2\) on the left. This can be done by subtracting \(2\) from both sides of the inequality:\[-3y + 2 - 2 < 17 - 2\] which simplifies to \[-3y < 15\].

Solve for the Variable

Next, you need to get \(y\) by itself. Divide both sides of the inequality by \(-3\) to solve for \(y\). Remember that dividing by a negative number reverses the inequality sign. So,\[y > \frac{15}{-3}\] which simplifies to \[y > -5\].

Verify the Solution

Double-check the result by substituting \(y = -4\) into the original inequality potential true case, ensuring it satisfies the condition. Thus, \(-3(-4) + 2 = 12 + 2 < 17\), which confirms the inequality holds.

Key concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is used to express several mathematical concepts in a general way. In algebra, you'll often work with letters like \(x\) or \(y\) to represent quantities that can vary. This allows us to write general formulas and solve problems for many different numbers. In the context of inequalities, algebra helps us express relationships where one quantity is not exactly equal to another, but instead is either less than or greater than. By using algebraic techniques, we can find the ranges of possible values that satisfy the inequality.

Solving Inequalities

Solving inequalities involves finding all the possible values of a variable that make the inequality true. Unlike equations that show equality, inequalities express a range of possibilities. Here is a step-by-step approach to solving inequalities:

• First, simplify the inequality if needed. This might involve distributing factors or combining like terms.
• Next, perform operations to both sides of the inequality to isolate the variable. This step is similar to solving equations.
• Pay attention to the inequality sign. Certain operations, such as multiplying or dividing by a negative number, require reversing the inequality sign.
• Once isolated, the variable should yield a range of values that constitutes the solution.

By solving inequalities, you can understand the extent of options or scenarios that meet a given condition. This is particularly useful in situations where definitive answers are not possible, but a spectrum of possibilities better describes the reality.

Variable Isolation

Variable isolation is a crucial technique in solving mathematical problems. The goal is to get the variable of interest alone on one side of the inequality or equation. This makes it easier to see what values the variable can take.

• Start by identifying the term involving the variable that needs to be isolated.
• Eliminate other terms by performing the inverse operations. For instance, if a term is added to the variable, subtract it from both sides.
• After removing constants or other terms, focus on the coefficient of the variable. You can divide or multiply through to simplify it to 1.

This process is like peeling away layers to reveal the variable, helping you understand its potential values easier. By isolating the variable, you can directly compare it to the value or range set by the other side of the inequality.

Inequality Sign Reversal

When solving inequalities, reversing the inequality sign is an important rule to remember. This rule applies specifically when you multiply or divide both sides of an inequality by a negative number.

• Initially, the inequality sign shows a particular relational order, such as greater than or less than.
• Multiplying or dividing by a negative number changes the direction of inequality because the relative order of numbers on a number line is reversed.
• For example, in the inequality \(-3y < 15\), dividing both sides by \(-3\) flips the sign, resulting in \(y > -5\).

Always being cautious of sign changes ensures that your solution remains true to the original inequality's meaning. It is one of the most common places errors can occur, so always double-check your work when negative multipliers or divisors are involved.