/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 86 Solve and graph. $$ \left|\f... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 86

Solve and graph. $$ \left|\frac{1+3 x}{5}\right|>\frac{7}{8} $$

Short Answer

Expert verified
x < -\frac{43}{24} or x > \frac{9}{8}

Step by step solution

01

- Remove the Absolute Value

The absolute value inequality \(\left|\frac{1+3x}{5}\right| > \frac{7}{8}\) can be split into two inequalities: \(\frac{1+3x}{5} > \frac{7}{8}\) and \(\frac{1+3x}{5} < -\frac{7}{8}\).
02

- Solve the First Inequality

To solve \(\frac{1+3x}{5} > \frac{7}{8}\), multiply both sides by 5 to get: \(1 + 3x > \frac{35}{8}\). Then, isolate x: \ 3x > \frac{35}{8} - 1 \3x > \frac{35}{8} - \frac{8}{8} \3x > \frac{27}{8} \ x > \frac{27}{24} \x > \frac{9}{8}\.
03

- Solve the Second Inequality

To solve \(\frac{1+3x}{5} < -\frac{7}{8}\), multiply both sides by 5 to get: \(1 + 3x < -\frac{35}{8}\). Then, isolate x: \3x < -\frac{35}{8} - 1 \3x < -\frac{35}{8} - \frac{8}{8} \3x < -\frac{43}{8} \ x < -\frac{43}{24} \x < -\frac{43}{24}\.
04

- Combine the Results

Combining the two solutions gives \(| {x < -\frac{43}{24}} \text{ or } {x > \frac{9}{8}}|\). This is because the original absolute value inequality splits into two separate ranges where the inequality holds true.
05

- Graph the Solution

On a number line, plot two open circles at \(-\frac{43}{24}\) and \(\frac{9}{8}\). Shade the region to the left of \(-\frac{43}{24}\) and the region to the right of \(\frac{9}{8}\). These represent the sets of values for x which satisfy the inequality.

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Key Concepts

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

Solving Inequalities
Inequalities help us understand ranges of values that a variable can take. Solving an inequality is similar to solving an equation. However, there are specific rules to follow for inequalities, especially when dealing with absolute values.
For the absolute value inequality \(\left|\frac{1+3x}{5}\right|\frac{7}{8}\right| \), we split it into two separate inequalities:
  • \(\frac{1+3x}{5} \frac{7}{8}\right| \)
  • \(\frac{1+3x}{5} \$ \frac{7}{8} \left\right.\right\right> -1 \)

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Most popular questions from this chapter

A \(\$ 6.00\) toll is charged to cross the bridge from mainland Florida to Sanibel Island. A six-month reduced-fare pass, costing \(\$ 50.00\) reduces the toll to \(\$ 2.00 .\) A six-month unlimited-trip pass costs \(\$ 300\) and allows free crossings. How many crossings in six months does it take for the reducedfare pass to be the more economical choice?

For \(f(x)=\frac{5-2 x}{4 x+3},\) find all \(x\) -values for which \(f(x) \geq 0\).

Use a graphing calculator to graph each function and find solutions of \(f(x)=0 .\) Then solve the inequalities \(f(x)<0\) and \(f(x)>0\). $$ f(x)=x-\sqrt{x}, x \geq 0 $$

Solve. $$ 5 x(x+1)(x-1)>0 $$

Solve using substitution or elimination. $$ \begin{aligned} &3 x-2 y=4\\\ &5 x-3 y=5 \end{aligned} $$
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