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Chapter 1: Problem 71

Use interval notation to express the solution set of each inequality. $$|4-5 x| \geq 24$$

Short Answer

Expert verified
The solution of the given inequality \(|4-5x|\geq 24\) is \((-鈭�, -4] \cup [5.6, 鈭�)\)

Step by step solution

01

Break down the absolute value inequality

Firstly, break down the absolute value inequality into two separate inequalities: \(4-5x \geq 24\) and \(4-5x \leq -24\)
02

Solving the inequalities

Next up, solve these two inequalities one at a time.\n For the inequality \(4-5x \geq 24\):\n Subtract \(4\) from both sides of the inequality, which gives \(-5x \geq 20\).\n Then, divide both sides by \(-5\) to solve for \(x\). Note that when dividing or multiplying an inequality by a negative number, the direction of the inequality sign changes. This gives \(x \leq -4\).\n\nFor the inequality \(4-5x \leq -24\):\n Subtract \(4\) from both sides of the inequality, which gives \(-5x \leq -28\).\n Again, divide both sides by \(-5\) with changing direction of the inequality sign. The result is \(x \geq 28/5=5.6\).
03

Interval notation

The final step, involves expressing the solution using interval notation. Interval notation is a method of writing down a set of numbers. In this case:\n For \(x \leq -4\) the interval notation is: \((-鈭�, -4]\).\n For \(x \geq 5.6\) the interval notation is: \([5.6, 鈭�)\).

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Key Concepts

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

Solving Inequalities
Absolute value inequalities can seem tricky at first, but breaking them down into simpler steps makes them more manageable. In this exercise, we explored how to tackle an absolute value inequality: \(|4-5x| \geq 24\). The absolute value involves considering both positive and negative scenarios, which leads us to create two separate inequalities:
  • \(4-5x \geq 24\)
  • \(4-5x \leq -24\)
In each case, solving the inequality involves isolating \(x\). For \(4-5x \geq 24\), we simplify down to \(x \leq -4\). Likewise, for \(4-5x \leq -24\), we find \(x \geq 5.6\). When solving inequalities:
  • Subtract or add terms on both sides to isolate terms with \(x\).
  • Remember that multiplying or dividing by a negative number reverses the inequality sign.
Representing Solutions with Interval Notation
Interval notation is a concise way to express the solution sets of inequalities. It's especially useful with compound solutions like our example.
  • The solution for \(x \leq -4\) is written as \((-鈭�, -4]\). Brackets indicate whether endpoints are included (closed bracket) or not (open bracket).
  • The solution for \(x \geq 5.6\) becomes \([5.6, 鈭�)\).
Notice how we've used round brackets for \(鈭瀄) since infinity isn't a specific number and can't be "included." Interval notation helps communicate solutions clearly by summarizing large sets of numbers efficiently.
Precalculus Problem Solving
Encountering inequalities with absolute values is common in precalculus, a foundational math course designed to prepare students for calculus. This type of problem is essential for developing logical reasoning and analytical skills.
  • Understanding inequalities and their representations paves the way for exploring more advanced mathematical concepts.
  • It builds familiarity with equations, graph interpretations, and mathematical proofs.
Mastering these skills in precalculus prepares students for calculus, where they'll encounter more complex problems. By solving absolute value inequalities and expressing the solutions in interval notation, learners solidify their problem-solving toolkit, which is crucial for future studies in advanced mathematics.

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Most popular questions from this chapter

Use the properties of inequalities to solve each inequality. Write answers using interval notation. $$x+4>3 x+16$$

The monthly revenue \(R\) for a product is given by \(R-420 x-2 x^{2},\) where \(x\) is the price in dollars of each unit produced. Find the interval in terms of \(x\) for which the monthly revenue is greater than zero.

Use the properties of inequalities to solve each inequality. Write answers using interval notation. $$-4(x-5)=2 x+15$$

The notation \(\left.f(x)\right|_{a} ^{b}\) is used to denote the difference \(f(b)-f(a) .\) That is, $$\left.f(x)\right|_{a} ^{b}=f(b)-f(a)$$ Evaluate \(\left.f(x)\right|_{0} ^{b}\) for the given function \(f\) and the indicated values of \(a\) and \(b\). $$f(x)=\sqrt{8-x} ;\left.f(x)\right|_{0} ^{8}$$

A boat was purchased for \(\$ 44,000 .\) Assuming the boat depreciates at a rate of \(\$ 4200\) per year (straight-line depreciation) for the first 8 years, write the value \(v\) of the boat as a function of the time \(t\) (measured in years) for \(0 \leq t \leq 8\).
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