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.
Part A:
To solve the equation ax - 5 = 18, we want to isolate x.
Adding 5 to both sides of the equation:
ax - 5 + 5 = 18 + 5
ax = 23
To isolate x, we divide both sides of the equation by a:
(ax) / a = 23 / a
x = 23/a
Therefore, the solution to the equation is x = 23/a.
Part B:
To solve the inequality ax - 5 > 18, we want to find the values of x that make the inequality true.
Adding 5 to both sides of the inequality:
ax - 5 + 5 > 18 + 5
ax > 23
To isolate x, 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 without changing the inequality. 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
Therefore, the solution to the inequality ax - 5 > 18 is x > 23/a if a > 0, and x < 23/a if a < 0.
Answer to Questions:
1. To represent all the solutions of the inequality ax - 5 > 18 on a number line, we would draw an open circle at 23/a (if a > 0) or a closed circle at 23/a (if a < 0), and shade the region to the right (if a > 0) or to the left (if a < 0) of the circle.
2. The difference between solving an equation and an inequality is that solving an equation finds the specific value for the variable that makes the equation true, while solving an inequality finds the range of values for the variable that makes the inequality true. In an equation, there is usually only one solution, while in an inequality, there can be infinitely many solutions or a range of solutions.
can you simplify the writing
Certainly! Here is the simplified writing:
Part A:
Equation: ax - 5 = 18
Solution:
1. Add 5 to both sides:
ax - 5 + 5 = 18 + 5
ax = 23
2. Divide both sides by a:
(ax) / a = 23 / a
x = 23/a
The solution to the equation is x = 23/a.
Part B:
Inequality: ax - 5 > 18
Solution:
1. Add 5 to both sides:
ax - 5 + 5 > 18 + 5
ax > 23
2. Consider the sign of a:
- If a > 0, we can divide both sides by a:
(ax) / a > 23 / a
x > 23/a
- If a < 0, we divide both sides by a and reverse the inequality sign:
(ax) / a < 23 / a
x < 23/a
The solution to the inequality is x > 23/a if a > 0, and x < 23/a if a < 0.
Answers to Questions:
1. To represent all the solutions of ax - 5 > 18 on a number line, mark an open circle at 23/a (if a > 0) or a closed circle at 23/a (if a < 0), and shade the region to the right (if a > 0) or left (if a < 0) of the circle.
2. The difference between solving an equation and an inequality is that an equation finds a specific value for the variable that satisfies the equation, while an inequality finds a range of values for the variable that fulfill the inequality. Equations usually have one solution, while inequalities can have infinitely many solutions or a range of solutions.
make the writing smaller
Part A:
Equation: ax - 5 = 18
Solution:
1. ax - 5 + 5 = 18 + 5
ax = 23
2. (ax) / a = 23 / a
x = 23/a
Part B:
Inequality: ax - 5 > 18
Solution:
1. ax - 5 + 5 > 18 + 5
ax > 23
2. - If a > 0: (ax) / a > 23 / a, x > 23/a
- If a < 0: (ax) / a < 23 / a, x < 23/a
Answers to Questions:
1. On a number line, mark an open circle at 23/a (if a > 0) or a closed circle at 23/a (if a < 0), and shade the region to the right (if a > 0) or left (if a < 0) of the circle.
2. Solving an equation finds a specific value for the variable, while solving an inequality finds a range of values that fulfill the inequality. Equations usually have one solution, while inequalities can have infinitely many or a range of solutions.
Part A:
To solve the equation ax - 5 = 18, we need to isolate the variable x by performing various operations. Follow the steps below:
Step 1: Add 5 to both sides of the equation to eliminate the constant term on the left side:
ax - 5 + 5 = 18 + 5
ax = 23
Step 2: Divide both sides of the equation by a to isolate x:
(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, we'll follow the same steps as in Part A, but with one additional consideration. Since we have an inequality symbol, if we divide or multiply both sides by a negative number, we need to reverse the inequality.
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:
(ax) / a > 23 / a
Now, we need to consider if a is positive or negative:
If a > 0:
x > 23/a
If a < 0:
x < 23/a
These are the solutions to the inequality ax - 5 > 18, depending on the sign of a.
Answer to Question 1:
To represent the solutions on a number line, draw a line with arrows on both ends indicating infinity. Then, mark the point corresponding to x = 23/a on the line, and shade the region to the right if a > 0 (x > 23/a) or to the left if a < 0 (x < 23/a). This represents all the solutions for the inequality ax - 5 > 18.
Answer to Question 2:
The main difference between solving an equation and an inequality is that when solving an equation, we want to find one specific value for the variable that makes the equation true. However, when solving an inequality, we want to find a range of possible values for the variable that satisfy the given inequality. Therefore, the solution to an inequality is expressed using inequality symbols (<, >, ≤, ≥) indicating the range of values that satisfy the inequality, while the solution to an equation is typically expressed as a single value.
Part A: Solve the equation ax - 5 = 18.
To solve the equation ax - 5 = 18, we need to isolate x. Here are the steps to solve it:
Step 1: Add 5 to both sides of the equation to get rid of the -5 term.
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: Solve the inequality ax - 5 > 18.
To solve the inequality ax - 5 > 18, we also need to isolate x. Here are the steps to solve it:
Step 1: Add 5 to both sides of the inequality to eliminate the -5 term.
ax - 5 + 5 > 18 + 5
ax > 23
Step 2: Divide both sides of the inequality by 'a'. However, we need to be cautious when dividing by 'a' in an inequality. If 'a' is positive, then the inequality does not change. If 'a' is negative, then the inequality switches direction. So we divide by 'a' while considering the sign of 'a'.
Case 1: If 'a' > 0, the inequality does not change.
ax > 23
x > 23/a
Case 2: If 'a' < 0, the inequality switches direction.
ax > 23
x < 23/a
Therefore, the solutions to the inequality ax - 5 > 18 are:
If 'a' > 0: x > 23/a
If 'a' < 0: x < 23/a
Answer to question 1:
To represent the solutions of the inequality ax - 5 > 18 on a number line, do the following:
- Mark a point on the number line representing 23/a.
- If 'a' is positive, shade the region to the right of the point.
- If 'a' is negative, shade the region to the left of the point.
This will show all the solutions for the inequality.
Answer to question 2:
The difference between solving an equation and an inequality lies in the solution set.
- In an equation, the solution is a specific value or set of values that satisfy the equation.
- In an inequality, the solution is a range of values, as it represents all the values that make the inequality true.