2.7 Solve Linear Inequalities

Before you get started, take this readiness quiz.

Translate from algebra to English: $15 > x$.
If you missed this problem, review the reference.

Solve: $n - 9 = −42 .$
If you missed this problem, review the reference.

Solve: $−5 p = −23 .$
If you missed this problem, review the reference.

Solve: $3 a - 12 = 7 a - 20 .$
If you missed this problem, review the reference.

Fighter pilots The maximum height, h, of a fighter pilot is 77 inches. Write this as an inequality.

Shopping The number of items, n, a shopper can have in the express check-out lane is at most 8. Write this as an inequality.

Give an example from your life using the phrase ‘at most’.

Explain why it is necessary to reverse the inequality when solving $\frac{n}{−3} < 12$.

Determine whether each number is a solution to the equation.

$w + 2 = \frac{5}{8} ; w = \frac{3}{8}$

Determine whether each number is a solution to the equation.

$6 a - 3 = −7 a , a = \frac{3}{13}$

Solve each equation using the Subtraction Property of Equality.

$y + 2 = −6$

Solve each equation using the Subtraction Property of Equality.

$n + 3.6 = 5.1$

Solve each equation using the Addition Property of Equality.

$x - 9 = −4$

Solve each equation using the Addition Property of Equality.

$p - 4.8 = 14$

Solve each equation.

$y + 16 = −9$

Solve each equation.

$d - 3.9 = 8.2$

Solve each equation.

$7 x + 10 - 6 x + 3 = 5$

Solve each equation.

$8 \left(3 p + 5\right) - 23 \left(p - 1\right) = 35$

Translate each English sentence into an algebraic equation and then solve it.

Four less than $n$ is 13.

Translate into an algebraic equation and solve.

Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh weigh?

Translate into an algebraic equation and solve.

Elissa earned $152.84 this week, which was $21.65 more than she earned last week. How much did she earn last week?

Solve each equation using the division and multiplication properties of equality and check the solution.

$13 a = −65$

Solve each equation using the division and multiplication properties of equality and check the solution.

$− y = 4$

Solve each equation using the division and multiplication properties of equality and check the solution.

$\frac{y}{−10} = 30$

Solve each equation using the division and multiplication properties of equality and check the solution.

$\frac{5}{8} u = \frac{15}{16}$

Solve each equation using the division and multiplication properties of equality and check the solution.

$\frac{c}{9} = 36$

Solve each equation using the division and multiplication properties of equality and check the solution.

$\frac{11}{12} = \frac{2}{3} y$

Solve each equation requiring simplification.

$24 x + 8 x - 11 x = −7 - 14$

Solve each equation requiring simplification.

$−9 \left(d - 2\right) - 15 = −24$

Translate to an equation and then solve.

The quotient of b and 9 is $−27$.

Translate to an equation and then solve.

The difference of s and one-twelfth is one fourth.

Translate into an equation and solve.

Janet gets paid $24 per hour. She heard that this is $\frac{3}{4}$ of what Adam is paid. How much is Adam paid per hour?

Solve the following equations with constants on both sides.

$10 w - 5 = 65$

Solve the following equations with constants on both sides.

$32 = −4 - 9 n$

Solve the following equations with variables on both sides.

$5 a + 21 = 2 a$

Solve the following equations with variables on both sides.

$4 x - \frac{3}{8} = 3 x$

Solve the following equations with variables and constants on both sides.

$5 n - 20 = −7 n - 80$

Solve the following equations with variables and constants on both sides.

$\frac{5}{8} c - 4 = \frac{3}{8} c + 4$

Solve each linear equation.

$9 \left(2 p - 5\right) = 72$

Solve each linear equation.

$8 + 3 \left(n - 9\right) = 17$

Solve each linear equation.

$\frac{1}{3} \left(6 m + 21\right) = m - 7$

Solve each linear equation.

$0.25 \left(q - 8\right) = 0.1 \left(q + 7\right)$

Solve each linear equation.

$5 + 7 \left(2 - 5 x\right) = 2 \left(9 x + 1\right)$
$- \left(13 x - 57\right)$

Solve each linear equation.

$2 \left[−16 + 5 \left(8 k - 6\right)\right]$
$= 8 \left(3 - 4 k\right) - 32$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$9 u + 32 = 15 \left(u - 4\right)$
$- 3 \left(2 u + 21\right)$

Classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.

$21 \left(c - 1\right) - 19 \left(c + 1\right)$
$= 2 \left(c - 20\right)$

Solve each equation with fraction coefficients.

$\frac{1}{3} x + \frac{1}{5} x = 8$

Solve each equation with fraction coefficients.

$\frac{1}{2} \left(k - 3\right) = \frac{1}{3} \left(k + 16\right)$

Solve each equation with fraction coefficients.

$\frac{5 y - 1}{3} + 4 = \frac{−8 y + 4}{6}$

Solve each equation with decimal coefficients.

$0.36 u + 2.55 = 0.41 u + 6.8$

Solve each equation with decimal coefficients.

$0.7 y + 2.5 = 0.95 y - 9.25$

Solve.

Mallory is taking the bus from St. Louis to Chicago. The distance is 300 miles and the bus travels at a steady rate of 60 miles per hour. How long will the bus ride be?

Solve.

Link rode his bike at a steady rate of 15 miles per hour for $2 \frac{1}{2}$ hours. How much distance did he travel?

Solve.

Use the formula. $d = r t$ to solve for $r$
ⓐ when when $d = 451$ and $t = 5.5$
ⓑ in general

Solve.

Use the formula $A = \frac{1}{2} b h$ to solve for $h$
ⓐ when $A = 153$ and $b = 18$
ⓑ in general

Solve.

Solve the formula $4 x + 3 y = 6$ for y
ⓐ when $x = −2$
ⓑ in general

Solve.

Solve the formula $V = L W H$ for $H$.

Determine whether each number is a solution to the equation $6 x - 3 = x + 20$.

ⓐ 5ⓑ $\frac{23}{5}$

Solve each equation.

$\frac{9}{2} c = 144$

Solve each equation.

$−8 x - 15 + 9 x - 1 = −21$

Solve each equation.

$\frac{2}{3} x = 6$

Solve each equation.

$10 y = −5 y - 60$

Solve each equation.

$9 m - 2 - 4 m - m = 42 - 8$

Solve each equation.

$− \left(d - 9\right) = 23$

Solve each equation.

$2 \left(6 x - 5\right) - 8 = −22$

Solve each equation.

$\frac{1}{4} p - \frac{1}{3} = \frac{1}{2}$

Solve each equation.

$14 n - 3 \left(4 n + 5\right) = −9 + 2 \left(n - 8\right)$

Solve each equation.

Solve the formula $x - 2 y = 5$ for y
ⓐ when $x = −3$
ⓑ in general

Translate to an equation or inequality and solve.

Fifteen more than n is at least 48.

Translate to an equation or inequality and solve.

Jenna bought a coat on sale for $120, which was $\frac{2}{3}$ of the original price. What was the original price of the coat?