Solve the equation and inequality for x.
Part A: Solve the equation showing all steps.
ax−5=18
Part B: Solve the inequality showing all steps.
ax−5>18
Answer these Questions:
1. Describe the number line set up that would show all of the solutions for the inequality in Part B.
2. State the difference between solving an equation and an inequality.
(4 points)
Part A:
To solve the equation ax - 5 = 18, we need to isolate the variable x.
First, let's add 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
ax = 23
Next, divide both sides of the equation by a:
(ax)/a = 23/a
x = 23/a
Therefore, the solution to the equation ax - 5 = 18 is x = 23/a.
Part B:
To solve the inequality ax - 5 > 18, we again need to isolate the variable x.
First, let's add 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
ax > 23
Next, divide both sides of the inequality by a. However, we need to be careful when dividing by a variable since we don't know if it's positive or negative.
If a > 0, then we can divide both sides by a without changing the direction of the inequality:
(ax)/a > 23/a
x > 23/a
If a < 0, then we need to divide both sides by a but flip the direction of the inequality:
(ax)/a < 23/a
x < 23/a
Therefore, the solution to the inequality ax - 5 > 18 depends on the sign of a. If a > 0, the solution is x > 23/a. If a < 0, the solution is x < 23/a.
Answer to Questions:
1. To show all the solutions for the inequality ax - 5 > 18 on a number line, you would first mark a point at 23/a (the value of x when a > 0) and shade to the right of that point to represent all values of x that satisfy the inequality. If a < 0, you would mark a point at 23/a and shade to the left of that point.
2. The main difference between solving an equation and an inequality is that an equation has only one solution, whereas an inequality can have multiple solutions. When solving an equation, we find the exact value for the variable that makes the equation true. When solving an inequality, we find a range of values for the variable that satisfy the inequality.
Part A: Solving the equation ax - 5 = 18
Step 1: Add 5 to both sides of the equation to isolate the variable:
ax - 5 + 5 = 18 + 5
ax = 23
Step 2: Divide both sides of the equation by "a" to solve for x:
(ax)/a = 23/a
x = 23/a
Therefore, the solution to the equation ax - 5 = 18 is x = 23/a.
Part B: Solving the inequality ax - 5 > 18
Step 1: Add 5 to both sides of the inequality to isolate the variable:
ax - 5 + 5 > 18 + 5
ax > 23
Step 2: Divide both sides of the inequality by "a" to solve for x:
(ax)/a > 23/a
x > 23/a
Therefore, the solution to the inequality ax - 5 > 18 is x > 23/a.
1. To represent the solutions for the inequality in Part B on a number line, you would start by marking a point at 23/a. Then, you would shade the area to the right of that point to represent all the possible values of x that satisfy the inequality x > 23/a.
2. The difference between solving an equation and solving an inequality is that when solving an equation, we are finding the specific value(s) of the variable that make the equation true. On the other hand, when solving an inequality, we are finding the range of values that make the inequality true. In an inequality solution, we often express the solution as a range or interval of possible values.