QuestionSolve $10|7 x+7|+10 \geq 80$ for $x$ .

Studdy Solution

STEP 1

Assumptions1. The absolute value of a number is always non-negative. . The inequality involves an absolute value, which means we need to consider two cases when the expression inside the absolute value is positive and when it is negative.

STEP 2

First, let's simplify the inequality by dividing each side by10.
$|7x +7| \geq8$

STEP 3

Now, we need to consider two cases, because the absolute value of a number is its distance from zero on the number line, which is always non-negative.
Case1 The expression inside the absolute value is non-negative, i.e., $7x +7 \geq0$ .
Case2 The expression inside the absolute value is negative, i.e., $-(7x +7) \geq0$ .

STEP 4

Let's solve for $x$ in Case1.
$7x +7 \geq8$

STEP 5

Subtract7 from both sides of the inequality.
$7x \geq1$

STEP 6

Divide both sides of the inequality by.
$x \geq \frac{1}{}$

STEP 7

Now, let's solve for $x$ in Case2.
$-(7x +7) \geq$

STEP 8

istribute the negative sign inside the parentheses.
$-7x -7 \geq8$

STEP 9

Add7 to both sides of the inequality.
$-7x \geq15$

STEP 10

Divide both sides of the inequality by -7. Remember that when you divide or multiply an inequality by a negative number, the direction of the inequality sign changes.
$x \leq -\frac{15}{7}$ So, the solution to the inequality $10|7x +7| +10 \geq80$ is $x \geq \frac{}{7}$ or $x \leq -\frac{15}{7}$ .

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