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Chapter 4: Problem 13

Determine whether the given numbers are solutions of the inequality. \(2 y+3<6-y\) a) 0 b) 1 c) \(-1\) d) 4

Short Answer

Expert verified
0 and -1 are solutions. 1 and 4 are not solutions.

Step by step solution

01

Rewrite the inequality and isolate y

Start with the given inequality: \[2y + 3 < 6 - y\] To isolate y, first add y to both sides: \[2y + y + 3 < 6\] This simplifies to: \[3y + 3 < 6\]
02

Solve for y

Now, subtract 3 from both sides: \[3y + 3 - 3 < 6 - 3\] This simplifies to: \[3y < 3\] Next, divide both sides by 3: \[y < 1\]
03

Verify each given number

We now need to check if the numbers satisfy the condition \(y < 1\):- For 0: \(0 < 1\) is true- For 1: \(1 < 1\) is false- For -1: \(-1 < 1\) is true- For 4: \(4 < 1\) is false

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Key Concepts

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

Solving Inequalities
Inequalities in algebra are like equations, but instead of showing equality, they show that one side is greater than or less than the other. In this exercise, we start with the inequality: $$2y + 3 < 6 - y$$. The goal is to find the values of y that make this inequality true.
We begin by isolating the variable y. This means we'll manipulate the inequality by combining like terms and moving terms from one side to the other.
To get y by itself:
  • First, we add y to both sides: $$2y + y + 3 < 6 - y + y$$
  • Next, simplify both sides to get: $$3y + 3 < 6$$
  • Finally, move the constant term (3) by subtracting it from both sides: $$3y + 3 - 3 < 6 - 3$$
  • Now we have: $$3y < 3$$
    This process of isolating y is critical, as it helps set up the inequality to easily solve for y.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions using basic algebraic rules. Here, it is used to solve inequalities.
Once we have: $$3y < 3$$ The next step is to solve for y. We divide both sides by the coefficient of y (which is 3): $$\frac{3y}{3} < \frac{3}{3}$$
This simplifies to: $$y < 1$$ .
So, algebraic manipulation has transformed our initial inequality into a very simple one: y must be less than 1.
This means that any number smaller than 1 makes the original inequality true.
Always remember to perform the same operation on both sides of an inequality, just as you would with an equation. That keeps the inequality balanced.
Verifying Solutions
In the final step, we verify if the given numbers are solutions of the inequality.
The inequality we have is : $$y < 1$$ . . This means only numbers less than 1 will be solutions.
Let's check each number:
  • For 0: We substitute y with 0: $$0 < 1$$ True. So, 0 is a solution.

  • For 1: Substitute y with 1:$$1 < 1$$ False. So, 1 is not a solution.

  • For -1: Substitute y with -1: $$-1 < 1$$True. So, -1 is a solution.

  • For 4: Substitute y with 4: $$4 < 1$$ False. So, 4 is not a solution.

Verifying solutions is all about plugging in the values and seeing if they satisfy the inequality. Remember to double-check your steps to avoid mistakes!

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