Problem 109 Solve and graph. $$ 4 m-8&gt... [FREE SOLUTION]

91影视

Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 4: Problem 109

Solve and graph. $$ 4 m-8>6 m+5 \text { or } 5 m-8<-2 $$

Short Answer

Expert verified

The solution is $m < 1.2$, and the graph is shaded to the left of 1.2 with an open circle at 1.2.

Step by step solution

Solve the First Inequality

Start by solving the inequality $4m - 8 > 6m + 5$. First, subtract $6m$ from both sides to get $4m - 6m - 8 > 5$. Simplify this to get $-2m - 8 > 5$. Next, add 8 to both sides to get $-2m > 13$. Finally, divide both sides by -2, and remember to reverse the inequality sign, resulting in $m < -6.5$.

Solve the Second Inequality

Next, solve the inequality $5m - 8 < -2$. First, add 8 to both sides to get $5m < 6$. Finally, divide both sides by 5 to get $m < 1.2$.

Combine the Solutions

The solution to the inequality is $m < -6.5$ or $m < 1.2$. Simplify this to just $m < 1.2$ since any number less than -6.5 is also included in $m < 1.2$.

Graph the Solution

On a number line, shade all numbers less than 1.2. This means shading everything to the left of 1.2 and using an open circle at 1.2 to represent that 1.2 is not 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

Solving inequalities is similar to solving equations, but with a few added rules. Here's how you do it step by step. First, look at each inequality separately. For example, in this exercise, we start with the inequality:

4m - 8 > 6m + 5

To solve it, follow these steps:

Subtract the same term from both sides (in this case, subtracting 6m)
Simplify what you have
Add or subtract any remaining constants to isolate the variable (here, it鈥檚 m)
If you multiply or divide by a negative number, reverse the inequality sign

Remember, the ultimate goal is to get the variable (m) by itself on one side of the inequality.

Graphing Inequalities

Once you've solved the inequalities, it's important to visualize your solution. To graph an inequality on a number line:

First, plot the critical value. For example, if the solution to the inequality is m < 1.2, place an open circle at 1.2. The open circle indicates that 1.2 is not included in the solution.
Shade the region of the number line where the inequality holds true. In our case, since m is less than 1.2, shade everything to the left of 1.2.

The graph helps to see all the possible values for the variable that make the inequality true.

Combining Solutions

When you have more than one inequality, you need to combine their solutions. This process helps you find the overall solution set. In the given exercise, we first solve the inequalities:

4m - 8 > 6m + 5

5m - 8 < -2

After solving, we get:

m < -6.5
m < 1.2

To combine the solutions, we take the more inclusive solution. Here, since m < -6.5 is also included in m < 1.2, the combined solution is simply m < 1.2. This means m takes any value less than 1.2.

Reversing Inequality Signs

Reversing inequality signs is crucial when dealing with inequalities, particularly when you multiply or divide by a negative number. For instance, if you have an inequality like:

-2m > 13

To solve for m, you need to divide both sides by -2. When you do this, the inequality sign reverses. This means:

m < -6.5

Always remember to flip the inequality sign whenever you multiply or divide by a negative number. Forgetting to do so will result in an incorrect solution. Pay close attention to this rule in every step when handling inequalities.

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91影视

Solve and graph. $$ 4 m-8>6 m+5 \text { or } 5 m-8<-2 $$

Short Answer

Step by step solution

Solve the First Inequality

Solve the Second Inequality

Combine the Solutions

Graph the Solution

Key Concepts

Solving Inequalities

Graphing Inequalities

Combining Solutions

Reversing Inequality Signs

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