Question

Solve the equation and inequality for x.

Part A: Solve the equation for x showing all steps for credit: ax−5=18
Part B: Solve the inequality for x showing all steps for credit: ax−5>18
Part C: State the difference between an equation and an inequality regarding solving steps and solutions.

Answer 1

TY :DD

Answer 2

Part A:

To solve the equation ax−5=18 for x, we need to isolate x on one side of the equation.

Starting with the given equation:
ax−5=18

We want to get rid of the -5 on the left side of the equation, so we add 5 to both sides:
ax−5+5=18+5
ax=23

To solve for x, we divide both sides of the equation by a:
(ax)/a = 23/a
x = 23/a

So the solution to the equation ax−5=18 is x = 23/a.


Part B:

To solve the inequality ax−5>18 for x, we need to isolate x on one side of the inequality, but this time we also need to pay attention to the direction of the inequality sign.

Starting with the given inequality:
ax−5>18

First, we add 5 to both sides of the inequality:
ax−5+5>18+5
ax>23

Now, we divide both sides of the inequality by a. However, we need to consider two cases separately depending on the value of a:

1) If a > 0, we can divide by a without changing the direction of the inequality:
(ax)/a > 23/a
x > 23/a

2) If a < 0, we need to flip the direction of the inequality when dividing by a:
(ax)/a < 23/a
x < 23/a

So, the solution to the inequality ax−5>18, depending on the value of a, is:
- If a > 0, x > 23/a
- If a < 0, x < 23/a


Part C:

The difference between an equation and an inequality lies in the number of possible solutions.

An equation has a single solution or a finite number of solutions that make the equation true. When solving an equation, the steps involve manipulating the equation to isolate the variable and find the specific value(s) that satisfy the equation.

On the other hand, an inequality has an infinite number of solutions since it represents a range of values. When solving an inequality, the steps involve determining the range of values for the variable that make the inequality true. The solution to an inequality is usually expressed in terms of intervals or inequalities.