Problem 41 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ -7 y+10 \leq-4 $$

Step-by-Step Solution

Answer

$$ Answer: The solution to the inequality is $$y \leq 2.$$

Step 1: Add 7y to both sides of the inequality

We have the inequality: $$-7y + 10 \leq -4.$$ Adding 7y to both sides, we get: $$10 \leq 7y - 4.$$

Step 2: Subtract 10 from both sides

Subtracting 10 from both sides of the inequality, we have: $$-14 \leq 7y - 14.$$

Step 3: Divide by -7 and flip the inequality sign

Now we will divide both sides by -7: $$\frac{-14}{-7} \geq \frac{7y - 14}{-7}.$$ This simplifies down to: $$2 \geq y.$$ So the solution to this inequality can be written as $$y \leq 2.$$

Key Concepts

Inequality Notation

Understanding inequality notation is crucial for solving mathematical problems that involve expressions indicating that one value is not necessarily equal to another but may be greater or lesser. In the given exercise, the notation \( -7y + 10 \leq -4 \) is used to signify that \( -7y + 10 \) is either less than or equal to \( -4 \). The symbol \( \leq \) stands for 'less than or equal to', while its counterpart, \( \geq \) means 'greater than or equal to'.

These symbols form the basis of expressing inequalities and are necessary for writing the final solution to an inequality problem. For example, when we conclude that \( y \leq 2 \), it means that the variable \( y \) holds all values up to and including 2, which is a direct application of inequality notation.

Algebraic Manipulation

Algebraic manipulation refers to the use of various algebraic operations to rearrange and simplify expressions or equations. The process typically involves operations such as addition, subtraction, multiplication, and division, as well as factoring and expanding.

In our step-by-step solution, we applied algebraic manipulation when we added \( 7y \) and subtracted 10 from both sides of the original inequality. This process is designed to isolate the variable on one side. The goal is to systematically transform the inequality into a simpler form that clearly identifies the range of possible values for the variable. Such an approach requires a careful application of algebraic rules to maintain the correct relationship between the sides of the inequality.

Properties of Inequalities

The properties of inequalities are the set of rules that dictate how inequalities react to different operations. It is important to note that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed, as seen in step 3 of our solution.

This reversal is a critical property that preserves the truth of the inequality. Without flipping the sign, the inequality would yield incorrect results. Another key property of inequalities is the transitive property, which means if \( a \leq b \) and \( b \leq c \) then \( a \leq c \). Understanding and applying these properties correctly is essential for finding the solution to inequality problems and preventing mistakes.