aInequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign.
This inequality tells us that all values greater than or equal to 4 will satisfy the inequality.
Below we demonstrate the inequality by graphing the solution set on a number line. Notice that x can equal 4, which we show with a closed circle on the number line.
b Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign.
This inequality tells us that all values greater than 412 will satisfy the inequality.
Below we demonstrate the inequality by graphing the solution set on a number line. Notice that x cannot equal 412, which we show with an open circle on the number line.
Test a value from each interval to see if it satisfies the original inequality.
Step 1
We will start by solving the related equation.
x^2=- 4x+5 ⇔ 1x^2+ 4x+( - 5)=0
We see above that a= 1, b= 4, and c= - 5. Let's substitute these values into the Quadratic Formula to solve the equation.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=- 2 ± 3
x_1=- 2 + 3
x_2=- 2 - 3
x_1=1
x_2=- 5
Step 2
The solutions of the related equation are - 5 and 1. Let's plot them on a number line. Since the original is not a strict inequality, the points will be closed.
Step 3
Finally, we must test a value from each interval to see if it satisfies the original inequality. Let's choose a value from the first interval, x ≤ - 5. For simplicity, we will choose x=- 6.
Since x=- 6 did not produce a true statement, the interval x ≤ - 5 is not part of the solution. Similarly, we can test the other two intervals.
Interval
Test Value
Statement
Is It Part of the Solution?
- 5 ≤ x ≤ 1
0
0 ≤ 5 ✓
Yes
x ≥ 1
2
4 ≰ - 3 *
No
We can now write the solution and show it on a number line.
- 5 ≤ x ≤ 1
d We are asked to solve the given absolute value inequality.
|2x-7|>31
To do this, we will create a compound inequality by removing the absolute value. In this case the solution set contains the numbers that make the distance between 2x and 7 greater than 31 in the positive direction or in the negative direction.
2x-7>31 or 2x-7< - 31
Let's isolate x in both of these cases before graphing the solution set.
This inequality tells us that all values less than - 12 will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the combination of the solution sets.
First Solution Set:& x>19
Second Solution Set:& x< - 12
Combined Solution Set:& x< - 12 or x>19
Graph
The graph of this inequality includes all values less than - 12 or greater than 19. We show this by keeping the endpoints open.
