olve 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 x.
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 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:
To solve the inequality ax - 5 > 18, we need to isolate x.
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 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 on a number line, we would mark a point at 23/a and draw an arrow to the right, as the inequality states that x should be greater than 23/a.
2. The main difference between solving an equation and an inequality is that when solving an equation, we are finding a specific value for the variable that makes the equation true. In contrast, when solving an inequality, we are looking for a range of values for the variable that make the inequality true.
make it more ssimple
Part A:
ax - 5 = 18
Add 5 to both sides:
ax = 23
Divide both sides by a:
x = 23/a
The solution to the equation is x = 23/a.
Part B:
ax - 5 > 18
Add 5 to both sides:
ax > 23
Divide both sides by a:
x > 23/a
The solution to the inequality is x > 23/a.
1. On a number line, mark a point at 23/a and draw an arrow to the right to represent all values of x that are greater than 23/a.
2. The main difference between solving an equation and an inequality is that when solving an equation, we find a specific value for the variable that satisfies the equation, whereas 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:
To solve the equation, we need to isolate the variable x.
First, we can add 5 to both sides of the equation to get:
ax - 5 + 5 = 18 + 5
This simplifies to:
ax = 23
Next, we divide both sides of the equation by a to solve for x:
ax/a = 23/a
This simplifies to:
x = 23/a
So, the solution to the equation is x = 23/a.
Part B: Solving the inequality ax-5 > 18:
To solve the inequality, we need to isolate the variable x.
First, we can add 5 to both sides of the inequality to get:
ax - 5 + 5 > 18 + 5
This simplifies to:
ax > 23
Next, we divide both sides of the inequality by a. However, we need to consider the sign of a:
If a is positive, we can divide by a without changing the direction of the inequality:
x > 23/a
If a is negative, we need to divide by a and flip the direction of the inequality:
x < 23/a
So, the solution to the inequality depends on the sign of a, and it can be expressed as either x > 23/a or x < 23/a.
Now, let's move on to the questions:
1. To represent the solutions of the inequality ax-5 > 18 on a number line, we would first plot a point at 23 with an open circle since it is not included in the solution. If a is positive, we would shade everything to the right of 23, indicating that all values greater than 23 are solutions. If a is negative, we would shade everything to the left of 23, indicating that all values less than 23 are solutions.
2. The main difference between solving an equation and an inequality is the inclusion of a range of solutions in an inequality. When solving an equation, we aim to find the specific value(s) of the variable that satisfies the equation. However, when solving an inequality, we are looking for a range of values of the variable that make the inequality true. This range can be represented by an inequality notation or as a solution set on a number line.