Problem 82 Solve and graph the solution set... [FREE SOLUTION]

91影视

Elementary Algebra

John Redden

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 2: Problem 82

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -10 x-3 \leq 17 \text { or } 20 x-6>-26 $$

Short Answer

Expert verified

The solution in interval notation is [-2, 鈭�).

Step by step solution

Solve the First Inequality

We are given \[-10x - 3 \leq 17.\] To solve for $x$, we first add 3 to both sides, giving us:\[-10x \leq 20.\] Next, we divide both sides by -10. Remember to flip the inequality sign because we are dividing by a negative number:\[x \geq -2.\]

Solve the Second Inequality

The second inequality is \[20x - 6 > -26.\]Add 6 to both sides to isolate the term with $x$:\[20x > -20.\]Now, divide both sides by 20:\[x > -1.\]

Combine the Inequalities

The first inequality, $x \geq -2$, represents all $x$ values that are greater than or equal to -2. The second inequality, $x > -1$, represents all $x$ values that are greater than -1. Since this is an "or" inequality, we take the union of both solutions, meaning we include all values from both solutions:$x \geq -2$ OR $x > -1$.

Graph the Solution Set

Draw a number line. To represent $x \geq -2$, draw a solid dot at -2 and shade to the right. For $x > -1$, draw an open dot at -1 (because -1 is not included) and also shade to the right. The entire region from $-2$ to infinity will be shaded.

Write the Interval Notation

In interval notation, the solution $x \geq -2$ OR $x > -1$ is written as:\[[-2, \infty).\]This interval includes all numbers greater than or equal to -2.

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Key Concepts

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

Solving Inequalities

Solving inequalities is a fundamental skill in algebra that helps us find a range of possible values for a variable. Inequalities use symbols such as $>$, $<$, $\geq$, and $\leq$ to show that one side is greater or smaller than the other.
To solve an inequality:

Perform arithmetic operations similar to those in equations鈥攁ddition, subtraction, multiplication, or division鈥攖o isolate the variable.
Remember to flip the inequality sign whenever you multiply or divide by a negative number. This is because dividing by a negative reverses the order of the inequality.
Solve each part of a compound inequality separately, if needed, then combine the solutions logically (using "and" or "or").

In this exercise, two inequalities were solved. For $-10x - 3 \leq 17$, we added 3, then divided by -10 (flipping the inequality) to get $x \geq -2$. In the second inequality, $20x - 6 > -26$, we added 6 and divided by 20 to get $x > -1$. Understanding each step ensures a strong grasp of solving inequalities!

Interval Notation

Once you solve an inequality, you express its solution using interval notation. This is a way to write subsets of the real number line.
Interval notation uses:

Brackets [ ] to include endpoints if the inequality is a greater than or equal to ($\geq$) or less than or equal to ($\leq$).
Parentheses ( ) to exclude endpoints when the inequality is strictly greater than (>) or less than (<).
The symbol $ \infty $ or $-\infty$ to signify unbounded ranges, remembering $\infty$ is never included, so it's always paired with a parenthesis.

In the provided solution, the combined inequalities $x \geq -2$ or $x > -1$ were expressed in interval notation as $[-2, \infty)$. This notation succinctly captures all numbers from -2 to infinity, with -2 included.

Graphing Solution Sets

Graphing solution sets visually represents the solutions of inequalities on a number line, helping to clarify and check your work. To graph an inequality:

Draw a number line with appropriate values marked.
Use a solid dot to include a number (for $\geq$ or $\leq$ inequalities) or an open dot to exclude it (for > or <).
Shade the area to the right for greater than solutions or to the left for less than solutions.

For the example here, we graph $x \geq -2$ with a solid dot at -2 and $x > -1$ with an open dot at -1, shading to the right to show all valid solutions. The graph makes it clear that all numbers from -2 onward are part of the solution set, visualizing the logic behind combining the inequalities.

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91影视

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -10 x-3 \leq 17 \text { or } 20 x-6>-26 $$

Short Answer

Step by step solution

Solve the First Inequality

Solve the Second Inequality

Combine the Inequalities

Graph the Solution Set

Write the Interval Notation

Key Concepts

Solving Inequalities

Interval Notation

Graphing Solution Sets

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