Solve the inequality and graph the solution on a real number line

In summary, the problem is to solve the inequality (3x - 5)/(x - 5) > 4 by subtracting 4 from both sides and combining terms on the left. The critical numbers are found where the numerator and denominator are both equal to zero.
  • #1
megacat8921
8
0
(3x - 5)/(x - 5) > 4

How does one complete this problem?
 
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  • #2
We are given to solve:

\(\displaystyle \frac{3x-5}{x-5}>4\)

Now, one might be tempted to multiply through by $x-5$ to clear the denominator on the right side, as we would with an equation. But with inequalities, we have to be mindful of the sign of a multiplicative quantity, so our best strategy here is to subtract 4 from both sides:

\(\displaystyle \frac{3x-5}{x-5}-4>0\)

Now, you need to combine terms on the left, and then determine your critical numbers, which are found anywhere the numerator AND denominator is zero. Can you proceed?
 
  • #3
Yes, thank you.
 

FAQ: Solve the inequality and graph the solution on a real number line

What is an inequality?

An inequality is a mathematical statement that compares two quantities using symbols such as <, >, ≤, or ≥. It represents a relationship between the values of the two quantities, where one is less than, greater than, less than or equal to, or greater than or equal to the other.

How do you solve an inequality?

To solve an inequality, you follow the same rules as solving an equation, with one exception: when you multiply or divide both sides of the inequality by a negative number, the direction of the inequality sign must be reversed. The solution is the set of values that satisfy the inequality.

What does it mean to graph the solution of an inequality?

Graphing the solution of an inequality means representing the set of values that satisfy the inequality on a number line. The solution can be shown as a shaded region on the number line, with the direction of the shading determined by the direction of the inequality sign.

What is a real number line?

A real number line is a horizontal line that represents all the real numbers, which include both positive and negative numbers, as well as zero. The numbers are evenly spaced on the line, with zero in the center and positive numbers increasing to the right and negative numbers decreasing to the left.

Can you provide an example of solving an inequality and graphing the solution on a real number line?

Yes, for example, if we have the inequality 2x + 3 < 9, we can solve it by subtracting 3 from both sides to get 2x < 6. Then, we divide both sides by 2 to get x < 3. The solution set is all real numbers less than 3, which can be graphed as a shaded region to the left of the number 3 on the real number line.

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