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Chapter 1: Problem 50

Solve each compound inequality. Write the solution set using interval notation and graph it. $$5+x>3-x \text { or } 2 x-3>x$$

Short Answer

Expert verified
The solution set is \((-1, \infty) \cup (3, \infty)\).

Step by step solution

01

Solve the first inequality

Consider the first inequality: \[5 + x > 3 - x\]. First, add \(x\) to both sides to combine like terms: \[5 + x + x > 3 - x + x\] which simplifies to \[5 + 2x > 3\]. Next, subtract 5 from both sides: \[2x > -2\]. Finally, divide both sides by 2: \[x > -1\].
02

Solve the second inequality

Consider the second inequality: \[2x - 3 > x\]. First, subtract \(x\) from both sides to combine like terms: \[2x - 3 - x > x - x\] which simplifies to \[x - 3 > 0\]. Next, add 3 to both sides: \[x > 3\].
03

Combine the solution sets using 'or'

Since the compound inequality uses 'or', combine the solution sets of each inequality. The first solution set is \(x > -1\), and the second solution set is \(x > 3\). The combined solution set is the union of these intervals: \((-1, \infty) \cup (3, \infty)\).
04

Write the solution set in interval notation

The solution set in interval notation is \((-1, \infty) \cup (3, \infty)\).
05

Graph the solution set

On a number line, shade the intervals \((-1, \infty)\) and \((3, \infty)\). This means that all numbers greater than -1 and all numbers greater than 3 are part of the solution set, with open circles at -1 and 3.

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Key Concepts

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

interval notation
Interval notation is a way of writing subsets of the real number line. It's particularly useful to express the solution sets of inequalities. There are two types of interval notation:
  • Closed intervals: Includes the endpoints. Represented with square brackets, like \[a, b\].
  • Open intervals: Excludes the endpoints. Represented with parentheses, like \(a, b\).
For example, \(-1, \infty\) denotes all real numbers greater than \-1 but not including \-1. Similarly, \(3, \infty\) represents all real numbers greater than 3. When we combine these two intervals using 'or,' we use the union symbol \(\cup\) to join them. So, our final interval notation is \((-1, \infty) \cup (3, \infty)\).
solving inequalities
Solving inequalities is similar to solving regular equations, but with a few additional rules. The goal is to isolate the variable. Consider these steps:
  • Combining like terms: Move terms involving the variable to one side of the inequality.
  • Manipulating the inequality: Use addition, subtraction, multiplication, or division to isolate the variable.
For example, take the inequality \(-2x > 4\). You'd divide both sides by \(-2\), but remember to reverse the inequality sign when dividing by a negative number. It becomes \x < -2\. In compound inequalities like \5 + x > 3 - x \text{or} \ 2x - 3 > x\, you solve each part separately. Then, combine solutions using 'or' or 'and.'
graphing inequalities
Graphing inequalities helps visualize solution sets. We often use a number line where:
  • Open circles: Indicate that endpoints are not included (ex: \(a, \infty\)).
  • Closed circles: Used for inequalities that include endpoints (ex: \[a, b\]).
For the inequality \x > -1\, you'd draw an open circle at \-1\ and shade all numbers to the right. For \x > 3\, an open circle at 3 with shading to the right. Combining these gives us \(-1, \infty) \cup (3, \infty)\). On the number line, this shows shading starting just after \-1\ and continuing beyond 3. This representation makes it easy to see where solutions lie.

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