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Chapter 4: Problem 11

Solve each inequality and graph the solution set. $$-4(3 y+2) \leq 28$$

Short Answer

Expert verified
The solution to the inequality is \( y \geq -3 \). In the graph, a filled circle should be placed at -3, and an arrow should point to the right, indicating that 'y' is greater than or equal to -3.

Step by step solution

01

Distribute the multiplication

Since there is a multiplication of a number with a bracket that includes the variable 'y', start solving the inequality by distributing the multiplication. Here, notice that the number is negative which should be considered while distributing. Therefore, the inequality -4(3y+2) 鈮� 28 becomes -12y -8 鈮� 28.
02

Isolate the 'y' term on one side

To further simplify the inequality and find the solution, isolate 'y'. This can be done by adding 8 to both sides of the inequality. This gives: -12y 鈮� 36.
03

Solve for 'y'

Now to solve for 'y' divide both sides of the inequality by -12. Remember that when you divide an inequality by a negative number, you should flip the inequality symbol. This gives answer as y 鈮� -3.
04

Draw the solution set on a number line

For the inequality y 鈮� -3, make a circle at -3 on a number line. Because the inequality is 'greater than or equal to', fill in the circle at -3. And because 'y' is greater than or equal to -3, draw an arrow to the right side of the number line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a crucial mathematical principle that allows you to multiply a number by a sum or a difference inside parentheses. It ensures that each term inside the parentheses gets multiplied by the number outside.
When dealing with inequalities like \(-4(3y + 2) \leq 28\), it is important to account for negative numbers. Here, \(-4\) needs to multiply each term inside the parentheses. Thus, \(-4 \times 3y = -12y\) and \(-4 \times 2 = -8\), resulting in the expression \(-12y - 8 \leq 28\).
Remember, the order and signs must be maintained while distributing, especially when negatives are involved, to avoid errors that could impact the solution.
Solution Sets
A solution set encompasses all the values that satisfy a given inequality. For the inequality \(y \geq -3\), the solution set includes \(-3\) and all numbers greater than \(-3\).
To isolate the variable effectively, each step should follow inverse operations. Here's how it worked in this problem:
  • Add 8 to both sides to counter subtract \(-8\).
  • Then divide by \(-12\) while flipping the inequality as the division involves a negative number.
This methodically leads to the solution set: \(y \geq -3\), demonstrating all values that make the inequality true.
Number Line Graphing
Graphing on a number line helps visualize the solution. Here's how you can do it for \(y \geq -3\):
  • Locate \(-3\) on the number line and place a circle or dot there. Since it's "greater than or equal to," the circle is filled.
  • Draw an arrow from \(-3\) extending to the right, indicating all possible values greater than \(-3\).
This graphical representation provides a clear visual idea of the solution set, making it easier to understand which values are included.

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Most popular questions from this chapter

Consider the following information: If a house was purchased for 105,000 dollar and is expected to appreciate 900 dollar per year, its value \(y\) after \(x\) years is given by the formula \(y=900 x+105,000 .\) Find the expected value of the house in 10 years.

$$\text { If } x=105, \text { evaluate } \frac{x^{500}-x^{499}}{x^{499}}$$

Perform the operations. $$5 x^{5} y^{10}-\left(2 x y^{2}\right)^{5}+(3 x)^{5} y^{10}$$

Perform the operation. Subtract \((-4 a+b)\) from \(\left(6 a^{2}+5 a-b\right)\)

Perform the operations. $$\left(4 c^{2}+3 c-2\right)+\left(3 c^{2}+4 c+2\right)$$
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