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Chapter 3: Problem 18

Determine whether the given value of the variable satisfies the inequality. $$5+3(a-7)>-12 ; \quad a=2$$

Short Answer

Expert verified
Yes, \( a = 2 \) satisfies the inequality.

Step by step solution

01

- Substitute the given value of the variable

Replace the variable \(a\) with the given value 2 in the inequality.\[ 5 + 3(2 - 7) > -12 \]
02

- Simplify inside the parentheses

Calculate \(2 - 7\):\[ 5 + 3(-5) > -12 \]
03

- Multiply by the coefficient

Multiply \(3\) by \(-5\):\[ 5 + (-15) > -12 \]
04

- Perform the addition

Add \(5\) and \(-15\):\[ -10 > -12 \]
05

- Interpret the result

Since \(-10 > -12\) is true, the value \(a = 2\) satisfies the inequality.

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Key Concepts

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

substituting values
When we need to determine if a specific value satisfies an inequality, substituting the value into the inequality is our first step. In this problem, we start by replacing the variable 'a' with the given number 2. This transforms the original expression from: \[5 + 3(a-7) > -12\]to:\[5 + 3(2-7) > -12\].Substituting values is like putting a number into a machine; you replace all instances of the variable with the given number. This helps us to see if the resulting statement holds true when simplified further.
simplifying expressions
After substituting the value, the next step is simplifying the expression. Simplification involves performing arithmetic operations such as addition, subtraction, multiplication, and division. In our example, we first handle the operation inside the parentheses:\[2 - 7 = -5\]This changes our expression to:\[5 + 3(-5) > -12\].Next, we perform the multiplication:\[3 \times (-5) = -15\], transforming our inequality to:\[5 + (-15) > -12\].Finally, we complete the addition to get:\[-10 > -12\]. By simplifying expressions step-by-step, we make it easier to judge whether or not the inequality holds true.
inequality interpretation
Interpreting inequalities involves understanding the relationship between quantities. In our case, after simplification, we get the inequality\[-10 > -12\]. To interpret this, we must consider what the inequality means: Is -10 greater than -12? Yes! It means that the expression on the left (after substitution and simplification) is indeed greater than the value on the right. Therefore, \(-10 > -12\) is true, confirming that the value we substituted for 'a' meets the requirements of the original inequality. This interpretation phase solidifies our result by logically validating the inequality.
algebraic solutions
Finding algebraic solutions involves following a structured approach to solve for variables or validate expressions. The problem-solving steps are:
  • Substitute the given value into the inequality.
  • Simplify the expression.
  • Multiply or divide as necessary.
  • Compare the simplified result with the inequality.
The result of \(-10 > -12\) from our process shows that there is no contradiction in the inequality given the variable value 'a = 2'. Thus, 'a = 2' is a solution to the inequality. By systematically following these steps, we can easily determine whether a given value satisfies more complex inequalities too.

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Most popular questions from this chapter

Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. A jogger's heart rate \(N\) (in beats per minute) increases as his or her speed \(s\) (in feet per second) increases. A physiologist collects data on a particular jogger and finds that the equation \(N=1.67 s+55\) seems to fit the data quite well. According to this model, what speed would produce a heart rate of 85 beats per minute?

Set up an inequality and solve it. Be sure to clearly label what the variable represents. The medium side of a triangle is \(2 \mathrm{cm}\) longer than the shortest side, and the longest side is twice as long as the shortest side. If the perimeter of the triangle is to be at least \(30 \mathrm{cm}\) and no more than \(50 \mathrm{cm},\) what is the range of values for the shortest side?

Write 83 as a product of prime factors. Write 83 as a product of prime factors.

Perform the indicated operations and simplify as completely as possible. $$x^{2}\left(x^{3}-4 x\right)-\left(2 x^{5}-4 x^{3}\right)$$

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