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Chapter 0: Problem 5

Express the solution set of the given inequality in interval notation and sketch its graph. $$ 7 x-2 \leq 9 x+3 $$

Short Answer

Expert verified
The solution set in interval notation is \(\left[-\frac{5}{2}, \infty \right)\).

Step by step solution

01

Isolate the Variable

Start by isolating the variable on one side of the inequality. Begin by subtracting \(7x\) from both sides:\[7x - 2 - 7x \leq 9x + 3 - 7x\]This simplifies to:\[-2 \leq 2x + 3\]
02

Move Constant Term

Next, move the constant term on the right to the left side by subtracting 3 from both sides:\[-2 - 3 \leq 2x + 3 - 3\]Simplify the inequality:\[-5 \leq 2x\]
03

Solve for the Variable

Now, solve for \(x\) by dividing both sides by 2:\[\frac{-5}{2} \leq x\]This can also be written as:\[x \geq -\frac{5}{2}\]
04

Express in Interval Notation

The solution represents all \(x\) values greater than or equal to \(-\frac{5}{2}\). Therefore, the interval notation is:\[\left[-\frac{5}{2}, \infty \right)\]
05

Sketch the Graph

On a number line, place a closed dot at \(-\frac{5}{2}\) to indicate that it is included in the solution set. Extend a line to the right towards infinity, showing that all values greater than \(-\frac{5}{2}\) are included in the solution set.

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Key Concepts

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

Interval Notation
Interval notation provides a concise way to express the solution set of an inequality. It uses brackets and parentheses to describe the range of values.
Here are some key points:
  • Square brackets \([ ]\) are used to include an endpoint in the set, known as a "closed" interval.
  • Parentheses \(( )\) are used to exclude an endpoint, indicating an "open" interval.
For example, the interval \([-\frac{5}{2}, \infty)\) means that all values from \(-\frac{5}{2}\) up to infinity are included, and \(-\frac{5}{2}\) itself is part of the set.
This format helps to visually understand the range of solutions without needing a detailed graph.Using interval notation becomes especially handy in mathematics for representing inequalities succinctly. It eliminates the need for lengthy descriptions and makes the solution set easy to comprehend at a glance.
Solving Linear Inequalities
Solving linear inequalities is similar to solving linear equations. However, instead of finding a single solution, we're pinpointing a range of values that satisfy the inequality.

To solve inequalities:
  • Perform the same operations on both sides as you would with an equation, such as adding, subtracting, multiplying, or dividing.
  • Be mindful that multiplying or dividing both sides by a negative number reverses the inequality symbol.
  • Keep isolating the variable on one side to find the solution range.
For example, in the given inequality \(7x - 2 \leq 9x + 3\), we subtracted \(7x\) and moved constants to isolate \(x\).
The final step was to divide by 2, leading to \(x \geq -\frac{5}{2}\).This method systematically provides the solution, ensuring accuracy in identifying the correct range of values.
Graphing Inequalities
Graphing inequalities offers a visual representation of the solution set on a number line. By plotting the inequality, we can see which values of \(x\) satisfy it.
Here鈥檚 how to graph an inequality:
  • Start by plotting the critical value. For example, use a closed dot at the endpoint if the inequality includes it (\(\leq \) or \(\geq \)). Use an open dot if it does not.
  • Draw a line or arrow to indicate the direction of the solution. The line extends towards infinity if representing values greater than the given number.
For the inequality \(x \geq -\frac{5}{2}\), place a closed dot at \(-\frac{5}{2}\) on the number line.
Then, draw a line extending to the right to show that all values greater than \(-\frac{5}{2}\) satisfy this inequality.Graphing helps easily verify solutions and provides clear insight into the nature and extent of possible values.

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Most popular questions from this chapter

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x^{2}-4 x+3 y^{2}=-2\)

Calculate each of the following without using a calculator. (a) \(\sin 570^{\circ}\) (b) \(\cos \frac{9 \pi}{2}\) (c) \(\cos \left(\frac{-13 \pi}{6}\right)\)

Sketch the graph of each equation. $$ x=y^{2}-3 $$

Graph the function \(f(x)=\cos x+\frac{1}{50} \sin 50 x\) using the windows given by the following ranges of \(x\) and \(y\). (a) \(-5 \leq x \leq 5,-1 \leq y \leq 1\) (b) \(-1 \leq x \leq 1,0.5 \leq y \leq 1.5\) (c) \(-0.1 \leq x \leq 0.1,0.9 \leq y \leq 1.1\) Indicate briefly which \((x, y)\) -window shows the true behavior of the function, and discuss reasons why the other \((x, y)\) -windows give results that look different. In this case, is it true that only one window gives the important behavior, or do we need more than one window to graphically communicate the behavior of this function?

Write \(F(x)=\sqrt{1+\sin ^{2} x}\) as the composite of four functions, \(f \circ g \circ h \circ k\).
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