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Chapter 8: Problem 130

Solve and graph. $$ 4 a-2 \leq a+1 \leq 3 a+4 $$

Short Answer

Expert verified
\[ -\frac{3}{2} \leq a \leq 1\]

Step by step solution

01

Solve the first inequality

First, solve the inequality \(4a - 2 \leq a + 1\). Subtract \(a\) from both sides to get:\[3a - 2 \leq 1\]Next, add 2 to both sides:\[3a \leq 3\]Finally, divide by 3:\[a \leq 1\]
02

Solve the second inequality

Now, solve the inequality \(a + 1 \leq 3a + 4\). Subtract \(a\) from both sides to get:\[1 \leq 2a + 4\]Next, subtract 4 from both sides:\[-3 \leq 2a\]Finally, divide by 2:\[ -\frac{3}{2} \leq a\]
03

Combine the inequalities

Combine the results from Step 1 and Step 2 to find the solution set:\[ -\frac{3}{2} \leq a \leq 1\]Thus, the solution to the compound inequality is \[-\frac{3}{2} \leq a \leq 1\].
04

Graph the solution

On a number line, represent the solution set by shading the region between \(-\frac{3}{2}\) and \(1\). Include closed circles at \(-\frac{3}{2}\) and \(1\) to indicate that these values are included in the solution.

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Key Concepts

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

Solving Inequalities
To start with solving inequalities, you must treat each inequality separately. An inequality might look similar to an equation, but instead of the equals sign, it uses symbols like <, >, 鈮�, or 鈮�.

For the given problem:
  • First Inequality: Solve this step-by-step just like an algebraic equation but keep the inequality sign in mind.
  • 4a - 2 鈮� a + 1: Subtract a from both sides. Now we have 3a - 2 鈮� 1.
  • Add 2 to both sides: 3a - 2 + 2 鈮� 1 + 2 simplifies to 3a 鈮� 3.
  • Divide by 3, hence a 鈮� 1.
  • Second Inequality: Let's repeat a similar approach here
  • a + 1 鈮� 3a + 4: Subtract a from both sides. We get 1 鈮� 2a + 4.
  • Subtract 4 from both sides: We get -3 鈮� 2a.
  • Divide by 2: Thus, -3/2 鈮� a.
Graphing Inequalities
After solving the inequalities, it's important to visualize the solution. Graphing helps you see the range of values that satisfy the inequality. Let鈥檚 take the solutions we found previously.
  • We have a 鈮� 1 and -3/2 鈮� a.
  • The combined range is -3/2 鈮� a 鈮� 1.
When graphing inequalities on a number line,
  • Use a line to represent all possible values between the boundaries.
  • Since the inequalities are 鈮� or 鈮�, use closed circles (filled dots) on the boundary points to show that these values are part of the solution set.
  • If an inequality were < or >, we would instead use open circles (hollow dots).
Number Line Representation
A number line is a simple visual tool to represent solutions. Here's how we do it:
  • Draw a horizontal line and mark points for critical values
  • For the given inequality, mark -3/2 and 1
  • Shade the region between these markers since this is the range of values that satisfies both inequalities
  • Place closed circles at -3/2 and 1 to show that these values are included in the solution
This representation visually indicates that any number between -3/2 and 1, including -3/2 and 1 themselves, is a solution to the compound inequality, making it clear and easy to understand.

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Most popular questions from this chapter

For \(g(x)=(x+3)(x-2)(x+1),\) find all \(x\) -values for which \(g(x)<0\).

Use interval notation to write each domain. The domain of \(f-g,\) if \(f(x)=\sqrt{x+3}\) and \(g(x)=\sqrt{2 x-1}\)

Classify each of the following statements as either true or false. To solve \(\frac{x+2}{x-3}<0\) using intervals, we divide the number line into the intervals \((-\infty,-2)\) and \((-2, \infty)\).

For \(F(x)=x^{3}-7 x^{2}+10 x,\) find all \(x\) -values for which \(F(x) \leq 0\).

On many graphing calculators, the TEST key provides access to inequality symbols, while the LOGIC option of that same key accesses the conjunction and and the disjunction or. Thus, if \(y_{1}=x>-2\) and \(y_{2}=x<4,\) Exercise 55 can be checked by forming the expression \(y_{3}=y_{1}\) and \(y_{2} .\) The interval(s) in the solution set appears as a horizontal line 1 unit above the \(x\) -axis. (Be careful to "deselect" \(y_{1}\) and \(y_{2}\) so that only \(y_{3}\) is drawn.) Use the Test key to check Exercises \(59,63,65,\) and 67
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