Question

Solve the inequality. Graph the solution. \(4-8 y \geq 12\)

Short answer

The solution to the inequality is \(y \leq -1\), and its graphical representation extends left from '-1' including '-1' on a number line.

Step-by-step solution

Write down the given inequality

The inequality to solve is \(4-8y \geq 12\).

Isolate the unknown

Subtract 4 from both sides of the inequality to isolate '-8y' on one side. This results in \(-8y \geq 12 - 4\) which simplifies to \(-8y \geq 8\).

Solve for y

Divide both sides of the inequality by '-8'. Remember the rule: when multiplying or dividing by a negative number, reverse the direction of the inequality. This yields \(y \leq 8/-8\), which simplifies to \(y \leq -1\).

Graph the solution

Plot the solution \(y \leq -1\) on a number line. Mark '-1' on the number line. The solution to the inequality is all real numbers less than or equal to '-1'. So place a filled dot on '-1' (to indicate that '-1' is included in the solution set) and draw a line extending to the left, indicating all numbers less than '-1'.

Key concepts

Algebraic Inequalities

Understanding algebraic inequalities is crucial for solving equations that depict real-world scenarios where amounts are not fixed but variable. An inequality indicates a comparative relationship between quantities. The most common symbols used are '<' (less than), '>' (greater than), '\text{\(warrow\) (less than or equal to)}', and '\text{\(earrow\) (greater than or equal to)}'. In the given exercise, the inequality is '4-8y \text{\(earrow\) \(12} \)'.

When you are solving an inequality like this, the first step is to isolate the variable 'y'. In our case, we subtract 4 from both sides to get '-8y \text{\(earrow\) \(8/\)}'. The next step involves dividing both sides by the coefficient of 'y', which is -8. A key concept to remember is that dividing or multiplying both sides of an inequality by a negative number flips the inequality symbol. So '-8y \text{\(earrow\) \(8/ \)}' becomes 'y \text{\(warrow\) \(-1/ \)}' upon dividing by -8. This final inequality indicates that 'y' is any number less than or equal to -1.

Graphing Solutions

Graphing is a visual representation that helps to better understand the solutions to inequalities. Once you have isolated and solved for the variable, you can represent the solution on a number line. In our exercise, the solution ' y \text{\(warrow\) \(-1/ \)}' means that 'y' includes -1 and every number less than -1.

To graph this, first draw a horizontal line to represent the number line and mark the numbers like on a ruler. The number '-1' should be clearly marked. Since '-1' is included in the solution set (because the inequality symbol includes 'equal to'), you place a filled-in circle on '-1' to show inclusivity. Then draw a line or arrow extending to the left from '-1' to represent all numbers less than -1. This is a fundamental part of graphing solutions and helps to visualize what the set of possible 'y' values looks like.

Inequality Number Line

The inequality number line is a valuable tool for visualizing and understanding the set of numbers that satisfy an inequality. It is a straightforward approach where a horizontal line represents all possible values that a variable can take. Specific rules must be followed for marking the solutions:

• If the variable value is part of the solution, use a filled dot.
• Use an open dot if the variable value is not part of the solution.
• Draw a line or arrow from the dot to show the range of values that are part of the solution, depending on the direction of the inequality.

For the inequality ' y \text{\(warrow\) \(-1/ \)}', '-1' is part of the solution, so we use a filled-in dot and draw a line extending to the left showing all numbers that are less than '-1'. In constrast, if the inequality was ' y \text{\(warrow\) (without an equal sign) $-1/'}', we would then use an open dot at '-1' and extend the line to the left, indicating that '-1' is not part of the solution set but every number less than '-1' is.