inequalities
–1 ≤ x – 3 ≤ 5
–2x – 5 < –2 or x – 3 < –10
Be careful of how a negative x-term is handled in the solving process. Show all math work arriving at the solutions.
•Show the solution sets written algebraically and as a union or intersection of intervals. Describe in words what the solution sets mean, and then display a simple line graph for each solution set. This is demonstrated in the Instructor Guidance in the left navigation toolbar, in your online course.
•Incorporate the following five math vocabulary words into your discussion. Use bold font to emphasize the words in your writing. Do not write definitions for the words; use them appropriately in sentences describing your math work.
◦Compound inequalities
◦And
◦Or
◦Intersection
◦Union
-2x - 5 < -2
-5 < 2x-2
-3 < 2x
-3/2 < x or, x > -3/2
do the others similarly.
To find the solutions for the given inequalities, let's first solve each inequality individually and then combine the results.
1) -1 ≤ x - 3 ≤ 5:
To isolate x, let's add 3 to all parts of the inequality:
-1 + 3 ≤ x - 3 + 3 ≤ 5 + 3
2 ≤ x ≤ 8
The solution set for this inequality is x is greater than or equal to 2 and less than or equal to 8.
2) -2x - 5 < -2 or x - 3 < -10:
Let's solve the first inequality:
-2x - 5 < -2
First, let's add 5 to both sides of the inequality:
-2x - 5 + 5 < -2 + 5
-2x < 3
Next, divide both sides of the inequality by -2, remembering to flip the inequality sign since we are dividing by a negative number:
(-2x)/-2 > 3/-2
x > -3/2
So the solution to the first inequality is x is greater than -3/2.
Now, let's solve the second inequality:
x - 3 < -10
Add 3 to both sides of the inequality:
x - 3 + 3 < -10 + 3
x < -7
So the solution to the second inequality is x is less than -7.
Now, let's combine the two solutions using "or" since we are looking for the values of x that satisfy either of the inequalities:
The solution set for the combined inequality is x is less than -7 or x is greater than -3/2.
In interval notation, this can be written as (-∞, -7) ∪ (-3/2, ∞).
Graphically, the line graph representation would show an open circle at -7 on the number line to indicate that -7 is not included in the solution set. Similarly, an open circle at -3/2 is shown, indicating that -3/2 is also not included. Then, a shaded region is displayed to represent all the values of x that satisfy the inequality.