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 can start by isolating x.
Adding 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
ax = 23
Next, we can divide both sides of the equation by a to solve for x:
ax/a = 23/a
x = 23/a
So the solution to the equation is x = 23/a.
Part B:
To solve the inequality ax - 5 > 18, we can start by isolating x.
Adding 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
ax > 23
Next, we can divide both sides of the inequality by a to solve for x. However, we have to be careful if a is negative, as dividing by a negative number would reverse the inequality.
If a > 0:
ax/a > 23/a
x > 23/a
If a < 0:
ax/a < 23/a
x < 23/a
So the solution to the inequality depends on whether a is positive or negative. If a > 0, the solution is x > 23/a. If a < 0, the solution is x < 23/a.
Answer to Questions:
1. To represent all solutions for the inequality in Part B on a number line, we would start by marking a point at 23/a. If a is positive, we would shade the area to the right of 23/a to represent x > 23/a. If a is negative, we would shade the area to the left of 23/a to represent x < 23/a.
2. The difference between solving an equation and an inequality is that an equation is looking for a specific value of the variable that makes the equation true, while an inequality is looking for a range of values of the variable that satisfy the inequality. In an inequality, we may have multiple solutions that satisfy the inequality and represent a range of values.
Part A: Solve the equation showing all steps:
ax - 5 = 18
Step 1: Add 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
ax = 23
Step 2: Divide both sides of the equation by a:
ax/a = 23/a
x = 23/a
So, the solution to the equation is x = 23/a.
Part B: Solve the inequality showing all steps:
ax - 5 > 18
Step 1: Add 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
ax > 23
Step 2: Divide both sides of the inequality by a. However, if a is negative, we need to reverse the inequality sign.
If a > 0:
ax/a > 23/a
x > 23/a
If a < 0:
ax/a < 23/a
x < 23/a
So, the solutions to the inequality are:
If a > 0: x > 23/a
If a < 0: x < 23/a
Answering your questions:
1. To show all the solutions for the inequality in Part B on a number line, you would represent the critical point x = 23/a with an open circle (since it doesn't equal 23/a) and shade the region to the right of the critical point if a > 0 or shade the region to the left of the critical point if a < 0.
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(s) that make the equation true, whereas when solving an inequality, we are finding the range of values of the variable(s) that make the inequality true. In an equation, there is a single solution or set of solutions. In an inequality, there is often a range of solutions.