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Chapter 6: Problem 27

Solve the inequality. $$\frac{2}{3} x+3 \geq 11$$

Short Answer

Expert verified
The solution to the inequality is \(x \geq 12\).

Step by step solution

01

Subtract 3 from both sides

To isolate the term containing \(x\) on one side of the inequality, subtract 3 from both sides. This gives \(\frac{2}{3} x \geq 11 - 3 = 8 \).
02

Multiply both sides by reciprocal of \( \frac{2}{3} \)

To solve for \(x\), multiply both sides of the inequality by the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \). This will cancel out the fraction in front of \(x de\). It gives \(x \geq \frac{3}{2} \times 8 \). Beware that reversing the inequality direction is not necessary in this step since multiplication was done with a positive number.
03

Compute the result

Multiplying \( \frac{3}{2} \) with 8 results in 12. Therefore, the solution to the inequality is \(x \geq 12\).\n

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation is a key skill in solving inequalities and equations. It involves rearranging and combining terms to simplify expressions or isolate a variable. When we solve inequalities, we often perform manipulations similar to those used in solving equations.
  • Our goal is to isolate the variable on one side of the inequality.
  • We start off by identifying terms that don't contain the variable and removing them from that side. For instance, in the given exercise, the term "+3" was subtracted from both sides.
  • Balancing is crucial when manipulating. Any operation performed on one side must also be done on the other to maintain the inequality.
Through algebraic manipulation, we transform the initial complex inequality into a simpler form where solving for the variable becomes straightforward.
Reciprocal
Understanding the concept of a reciprocal is essential when dealing with fractions in equations or inequalities. The reciprocal of a number is simply 1 divided by that number.
  • For example, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \).
  • Reciprocals are used to eliminate fractions by multiplying both sides of an equation or inequality by the reciprocal of the fraction.
In the exercise, multiplying both sides by \( \frac{3}{2} \) effectively cancels out the fraction in front of \(x\), making it easier to solve. It's important to remember that multiplying or dividing by a positive reciprocal keeps the inequality direction unchanged. This move helps to cleanly isolate the variable, facilitating our path to the solution.
Isolation of Variable
The isolation of a variable is often the end goal in solving equations and inequalities. By isolating the variable, we find the value or range of values that makes the inequality true.
  • To isolate a variable, we perform operations that cancel out other terms or coefficients.
  • Starting from the inequality \( \frac{2}{3}x + 3 \geq 11 \), we aimed to leave \(x\) by itself on one side of the inequality.
  • This involved subtracting 3, then multiplying by \( \frac{3}{2} \), clearing the fraction.
Each step in isolating the variable must keep the balance of the inequality intact, using careful arithmetic and the reciprocal mastery. Once \(x\) is isolated, the problem is essentially solved, and we interpret the solution \(x \geq 12\) as the range for \(x\) values that satisfy the original inequality.

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Most popular questions from this chapter

Check whether \((0,0)\) is a solution. Then sketch the graph of the inequality. $$ x+y>3 $$

Check whether \((0,0)\) is a solution. Then sketch the graph of the inequality. $$ x>-2 $$

Sketch the graph of the inequality. $$y-x \leq 11$$

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