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Solve the compound inequality. Write the solutions in interval notation.
x + 7 ≥ 4 and 11x - 8 ≥ 3

User Razemauze
by
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2 Answers

23 votes
23 votes

Answer: [1, infinity)

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Step-by-step explanation:

Let's isolate x in the first inequality mentioned.

x + 7 ≥ 4

x + 7-7 ≥ 4-7

x ≥ -3

I subtracted 7 from both sides to undo the +7.

Now isolate x in the second inequality. We first add 8 to both sides, then divide both sides by 11.

11x - 8 ≥ 3

11x - 8+8 ≥ 3+8

11x ≥ 11

11x/11 ≥ 11/11

x ≥ 1

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The two solutions we found were

x ≥ -3 and x ≥ 1

Next, draw out a number line with closed circles at -3 and 1. Shade to the right of each closed circle endpoint as you see below.

Notice that the two inequalities overlap for x ≥ 1

In other words, if a number is -3 or larger AND 1 or larger, then the number is 1 or larger.

The two solutions intersect to form x ≥ 1

The inequality x ≥ 1 is the same as 1 ≤ x which is the same as 1 ≤ x < infinity

Then that turns into the interval notation of [1, infinity)

The square bracket includes the endpoint 1.

If needed, use the infinity symbol in place of the word "infinity".

NO LINKS!! Solve the compound inequality. Write the solutions in interval notation-example-1
User Jakebman
by
2.9k points
27 votes
27 votes

Answer:

[1, ∞)

Step-by-step explanation:

To solve a compound inequality, first separate it into two inequalities and solve them separately:

Solution of Inequality 1:

x + 7 ≥ 4

⇒ x ≥ 4 - 7

x ≥ -3

Solution of Inequality 2:

11x - 8 ≥ 3

⇒ 11x ≥ 3 + 8

⇒ 11x ≥ 11

x ≥ 1

As both graphs overlap for x ≥ 1, this is the solution.

Therefore, in interval notation: [1, ∞)

User Andebauchery
by
2.5k points