Think for which value will the inequality become an equality. Then, analyze nearby values to see what happens.
D
Practice makes perfect
We need to find the correct graph for the given inequality.
-3 < - xThe right-hand side of the inequality uses - x. This means that it is involving the opposites of the values for x shown in the graphs. Notice that the inequality would become an equality when x = 3, since then - x =- 3. Let's then analyze some values near 3 to see when the inequality holds.
x
- x
-3 ? < - x
1
-1
✓
2
-2
✓
3
-3
*
4
-4
*
5
-5
*
We can see that the inequality only holds for values of x that are less than 3. The graph which shades all numbers less than 3 is option D.
Alternative Solution
Manipulating the inequality
It would be easier to have an inequality in terms of x instead of - x. We can work with inequalities similarly as we do with equations, but there is an important difference. Think of the statement
5 > - 10. This is a true statement since 5 is greater than -10. If we add the same quantity to both sides or multiply by any positive number the statement will still be true.
5 + 3 > 10 +3 ⇒ 8 > -7
3(5) > 3(- 10) ⇒ 15 > - 30
Let's now try multiplying by a negative number. -3(5) ? > -3(-10) ⇒ - 15 < 30.
Did you see what happened? When we multiplied by a negative number both sides, what was negative became positive, and what was positive became negative. What was larger became smaller, and what was smaller became larger. Everything reversed. Hence, when we multiply by a negative number we must reverse the inequality sign. With this in mind, we can rewrite the exercise's inequality .
