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Chapter 1: Problem 3

Solve the inequality and mark the solution set on a number line. $$16 x+64 \leq 16$$.

Short Answer

Expert verified
The solution to the inequality is \(x \leq -3\). Therefore, any number less than or equal to -3 is part of the solution set.

Step by step solution

01

Simplify the inequality

Begin by simplifying the inequality. The goal is to isolate \(x\). The inequality is \(16x + 64 \leq 16\). Subtract 64 from both sides to isolate the term with \(x\). This results in \(16x \leq 16 - 64\)
02

Solve for \(x\)

After subtraction, your new inequality is \(16x \leq -48\). Divide each side by 16 to solve for \(x\). This results in \(x \leq -48/16\)
03

Simplify the answer

Simplify \(x \leq -48/16\) by dividing -48 by 16. This yields \(x \leq -3\)
04

Mark the solution set on the number line

Draw a number line and mark the point -3. Use a closed dot to show that -3 is included in the solution set. Draw an arrow to the left of -3 on the number line to show that any value less than or equal to -3 is a solution to the inequality.

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Key Concepts

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

Number Line Representation
When solving inequalities, it's crucial to visually represent the solution. The number line is an excellent tool for this purpose.
It allows us to see which numbers satisfy the inequality. For the inequality \(x \leq -3\), here's how to represent it on a number line:
  • First, draw a horizontal line to represent the number line.
  • Identify and mark the point -3 on the line. Since the inequality includes "less than or equal to," we use a closed dot at -3.
  • Place an arrow extending to the left from -3 to indicate all numbers less than -3 are included in the solution.
This closed dot at -3 means that -3 itself is included in the solution set. Anything to the left of -3 also satisfies the inequality. This visual representation makes it clear which values of \(x\) fulfill the inequality.
Simplification of Inequalities
Simplifying inequalities is a critical step in solving them. It usually involves basic algebraic operations to isolate the variable. Let鈥檚 look at how this works for the inequality \(16x + 64 \leq 16\):
  • Start by subtracting 64 from both sides to focus on the term with \(x\). This gives \(16x \leq 16 - 64\).
  • Calculate the right side: \(16 - 64 = -48\).
  • You now have the inequality \(16x \leq -48\).
The next step is to isolate \(x\). Since \(x\) is multiplied by 16, divide everything by 16:
  • This results in \(x \leq \frac{-48}{16}\).
  • Further simplify \(\frac{-48}{16}\) to get \(x \leq -3\).
Simplifying inequalities involves rearranging terms and arithmetic operations. The goal is to express \(x\) in terms of a simpler inequality.
Mathematical Problem Solving
Mathematical problem solving requires a systematic approach. It鈥檚 about understanding the problem, planning a solution, executing the plan, and reviewing the result. In our case of solving the inequality, these phases work as follows:
  • **Understanding:** Recognize the given inequality \(16x + 64 \leq 16\) and know you need to find values for \(x\).
  • **Planning:** Decide on using algebraic steps to isolate \(x\). You simplify and solve step by step.
  • **Executing:** Carry out each step carefully. Subtract constants, divide by coefficients, and simplify where possible.
  • **Reviewing:** After solving the problem, check your steps and verify the solution. Ensure that when \(x = -3\) and any number less than -3 is substituted back, the inequality holds.
This structured method ensures you don't miss steps and that the solution satisfies the original inequality. These problem-solving stages help tackle not just inequalities but a variety of math problems systematically.

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Most popular questions from this chapter

Sketch the graph of the function. $$f(x)=\sqrt{\sin ^{2} x}$$.

Form the compositions \(f \circ g\) and \(g \circ f,\) and specify the domain of each of these combinations. $$f(x)=x^{2}-2 x, \quad g(x)=x+1$$

Give the domain and range of the function. $$f(x)=\frac{1}{2} \sqrt{1-4 x^{2}}$$

Form the composition \(f \circ g\) and give the domain. $$f(x)=1 /(x-1), \quad g(x)=x^{2}$$

Form the composition \(f \circ g \circ h\) and give the domain. $$f(x)=\frac{x+1}{x}, \quad g(x)=\frac{1}{2 x+1}, \quad h(x)=x^{2}$$
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