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Chapter 17: Problem 30

Solve the inequalities by displaying the solutions on a calculator. See Examples 9 and 10. $$40(x-2)>x+60$$

Short Answer

Expert verified
The solution is \(x > 3.59\).

Step by step solution

01

Simplify the Inequality

Start by expanding and simplifying the inequality. Distribute the 40 to both terms inside the parenthesis: \[ 40(x-2) > x + 60 \] This becomes: \[ 40x - 80 > x + 60 \]
02

Move Terms Involving x

Subtract \(x\) from both sides to get terms containing \(x\) on one side:\[ 40x - x - 80 > 60 \] This simplifies to: \[ 39x - 80 > 60 \]
03

Isolate the Variable x

To isolate \(x\), add 80 to both sides:\[ 39x - 80 + 80 > 60 + 80 \] This leads to: \[ 39x > 140 \]
04

Solve for x

Divide both sides of the inequality by 39 to solve for \(x\): \[ x > \frac{140}{39} \] This simplifies to: \[ x > \frac{140}{39} \approx 3.59 \]
05

Verify with Calculator

Check this inequality on a calculator by choosing values greater than 3.59 to confirm they satisfy the inequality. For example, use \(x = 4\): \[ 40(4-2) > 4 + 60 \] \[ 80 > 64 \] The statement is true, confirming the solution.

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Key Concepts

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

Understanding Solving Inequalities
When you face an inequality like \(40(x-2)>x+60\), the goal is to determine a range of values that satisfy it. This involves manipulating and simplifying the inequality similar to solving equations. However, instead of seeking a single solution, we look for a set of solutions. The main difference lies in the outcome: inequalities give us a spectrum of numbers instead of one.
To tackle the problem, break it down step by step:
  • First, expand and simplify both sides of the inequality. This may involve distributing coefficients and combining like terms.
  • Next, isolate the variable of interest. Our goal is to get "x" alone on one side by performing operations like addition, subtraction, multiplication, or division.
  • Always pay attention to the inequality sign. If you multiply or divide by a negative number, the inequality sign flips direction.
Solving inequalities often results in expressions like \(x > a\) or \(x < b\), indicating that any number greater than \(a\) or less than \(b\) is part of the solution.
Mastering Inequality Simplification
Simplification is a crucial step in solving inequalities effectively. By simplifying the given inequality, you make the expression easier to handle and solve. Take, for instance, the equation \(40(x-2) > x + 60\), where our first move is to expand and simplify:
  • Distribute the 40 to yield \(40x - 80 > x + 60\).
  • Next, simplify by moving terms involving \(x\) to one side. Subtract \(x\) from both sides to get \(39x - 80 > 60\).
  • Finally, simplify further by moving constant terms to the other side. Add 80 to both sides, resulting in \(39x > 140\).
  • Finish by dividing both sides by 39, isolating \(x\), and arriving at \(x > \frac{140}{39}\), a more digestible format of the inequality.
Thanks to simplification, you can now easily determine the solution set that meets the inequality.
Embracing Mathematical Problem Solving
Mathematical problem-solving is about understanding the process, not just finding the answers. When approaching inequalities, this requires a methodical strategy to analyze and simplify problems.
Start by comprehending the basic concept of inequalities and the operations that apply to them. Inequalities can be manipulated similarly to equations, but you must consider the direction and behavior of the inequality sign.
Employ logical reasoning in each step: join calculations, check signs, and re-evaluate the simplified results. Here鈥檚 why it鈥檚 important:
  • Logical flow: Each step follows logically from the previous one. This helps in understanding the progression of the solution.
  • Simple checks: Verifying solutions, like substituting \(x = 4\) in the final step, acts as a check to make sure your solution satisfies the original inequality.
  • Adaptation to new problems: Once you understand the basics of inequality solving and problem simplification, you can apply these techniques to more complex or differently structured inequalities.
Embracing this problem-solving mindset turns inequality challenges into solvable puzzles.

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Most popular questions from this chapter

Set up the necessary inequalities and sketch the graph of the region in which the points satisfy the indicated system of inequalities. A rectangular computer chip is being designed such that its perimeter is no more than \(15 \mathrm{mm}\), its width at least \(2 \mathrm{mm}\) and its length at least \(3 \mathrm{mm}\). Graph the possible values of the width \(w\) and the length \(l\)

some applications of inequalities are shown. The velocity \(v\) of an ultrasound wave in soft human tissue may be represented as \(1550 \pm 60 \mathrm{m} / \mathrm{s}\), where the \(\pm 60 \mathrm{m} / \mathrm{s}\) gives the possible variation in the velocity. Express the possible velocities by an inequality.

Use inequalities involving absolute values to solve the given problems. The Mach number \(M\) of a moving object is the ratio of its velocity \(v\) to the velocity of sound \(v_{s},\) and \(v_{s}\) varies with temperature. A jet traveling at \(1650 \mathrm{km} / \mathrm{h}\) changes its altitude from \(500 \mathrm{m}\) to \(5500 \mathrm{m}\) At \(\left.500 \mathrm{m} \text { (with the temperature at } 27^{\circ} \mathrm{C}\right), v_{s}=1250 \mathrm{km} / \mathrm{h},\) and at \(5500 \mathrm{m}\left(-3^{\circ} \mathrm{C}\right), v_{s}=1180 \mathrm{km} / \mathrm{h} .\) Express the range of \(M,\) using an inequality with absolute values.

Set up the necessary inequalities and sketch the graph of the region in which the points satisfy the indicated system of inequalities. The cross-sectional area \(A\) (in \(\mathrm{m}^{2}\) ) of a certain trapezoid culvert in terms of its depth \(d\) (in \(\mathrm{m}\) ) is \(A=2 d+d^{2}\). Graph the possible values of \(d\) and \(A\) if \(A\) is between \(1 \mathrm{m}^{2}\) and \(2 \mathrm{m}^{2}\).

Solve the given problems by setting up and solving appropriate inequalities. Graph each solution. The mass \(m\) (in \(\mathrm{g}\) ) of silver plate on a dish is increased by electroplating. The mass of silver on the plate is given by \(m=125+15.0 t, \text { where } t \text { is the time (in } \mathrm{h})\) of electroplating. For what values of \(t\) is \(m\) between \(131 \mathrm{g}\) and \(164 \mathrm{g} ?\)
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